fact stringlengths 28 3.53k | type stringclasses 8 values | library stringclasses 3 values | imports listlengths 1 7 | filename stringlengths 33 87 | symbolic_name stringlengths 1 87 | docstring stringlengths 17 494 ⌀ |
|---|---|---|---|---|---|---|
nonempty_cardSet : ∀ n ≥ 3, (cardSet n).Nonempty := sorry /-- Depending on details of definitions, the statement is false or trivial for $n < 3$. -/ @[category test, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/107.lean | nonempty_cardSet | /-- For every $n ≥ 3$, there exists $N$ such that any $N$ points, no three on a line,
contain a convex $n$-gon. -/ |
f_zero_eq : f 0 = 0 := by have : ∀ P, HasConvexNGon 0 P := by intro; use ∅; simp [ConvexIndep] simp [f, cardSet, this] @[category test, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/107.lean | f_zero_eq | /-- Depending on details of definitions,
the statement is false or trivial for $n < 3$. -/ |
f_three_eq : f 3 = 3 := by sorry namespace variants /-- Erdős and Szekeres proved the bounds $$ 2^{n-2} + 1 ≤ f(n) ≤ \binom{2n-4}{n-2} + 1 $$ ([ErSz60] and [ErSz35] respectively). [ErSz60] Erdős, P. and Szekeres, G., _On some extremum problems in elementary geometry_. Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1960/61), 53-62. [ErSz35] Erdős, P. and Szekeres, G., _A combinatorial problem in geometry_. Compos. Math. (1935), 463-470. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/107.lean | f_three_eq | /-- Depending on details of definitions,
the statement is false or trivial for $n < 3$. -/ |
ersz_bounds : ∀ n ≥ 3, 2^(n - 2) + 1 ≤ f n ∧ f n ≤ Nat.choose (2 * n - 4) (n - 2) + 1 := by sorry /-- Suk [Su17] proved $$ f(n) ≤ 2^{(1+o(1))n}. $$ [Su17] Suk, Andrew, _On the Erdős-Szekeres convex polygon problem_. J. Amer. Math. Soc. (2017), 1047-1053. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/107.lean | ersz_bounds | /--
Erdős and Szekeres proved the bounds
$$
2^{n-2} + 1 ≤ f(n) ≤ \binom{2n-4}{n-2} + 1
$$
([ErSz60] and [ErSz35] respectively).
[ErSz60] Erdős, P. and Szekeres, G., _On some extremum problems in elementary geometry_.
Ann. Univ. Sci. Budapest. Eötvös Sect. Math. (1960/61), 53-62.
[ErSz35] Erdős, P. and Szekeres, G., _A combinatorial problem in geometry_.
Compos. Math. (1935), 463-470.
-/ |
su_bound : ∃ r : ℕ → ℝ, r =o[atTop] (fun n => (n : ℝ)) ∧ ∀ n ≥ 3, (f n : ℝ) ≤ 2^(n + r n) := by sorry /-- The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20], who prove $$ f(n) ≤ 2^{n+O(\sqrt{n\log n})}. $$ [HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János and Tardos, Gábor, _Two extensions of the Erdős-Szekeres problem_. J. Eur. Math. Soc. (JEMS) (2020), 3981-3995. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/107.lean | su_bound | /--
Suk [Su17] proved
$$
f(n) ≤ 2^{(1+o(1))n}.
$$
[Su17] Suk, Andrew, _On the Erdős-Szekeres convex polygon problem_.
J. Amer. Math. Soc. (2017), 1047-1053.
-/ |
hmpt_bound : ∃ r : ℕ → ℝ, r =O[atTop] (fun n => Real.sqrt (n * Real.log n)) ∧ ∀ n ≥ 3, (f n : ℝ) ≤ 2^(n + r n) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/107.lean | hmpt_bound | /--
The current best bound is due to Holmsen, Mojarrad, Pach, and Tardos [HMPT20],
who prove
$$
f(n) ≤ 2^{n+O(\sqrt{n\log n})}.
$$
[HMPT20] Holmsen, Andreas F. and Mojarrad, Hossein Nassajian and Pach, János and Tardos, Gábor,
_Two extensions of the Erdős-Szekeres problem_. J. Eur. Math. Soc. (JEMS) (2020), 3981-3995.
-/ |
f (p : ℕ) : ℕ := sInf {n | (n)! + 1 ≡ 0 [MOD p]} /-- Is it true that there are infinitely many $p$ for which $f(p) = p − 1$? -/ @[category research open, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1072.lean | f | /-- For any prime $p$, let $f(p)$ be the least integer such that $f(p)! + 1 \equiv 0 \mod p$.-/ |
erdos_1072a : answer(sorry) ↔ Set.Infinite {p | p.Prime ∧ f p = p - 1} := by sorry /-- Is it true that $f(p)/p \to 0$ for $p \to \infty$ in a density 1 subset of the primes? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1072.lean | erdos_1072a | /-- Is it true that there are infinitely many $p$ for which $f(p) = p − 1$? -/ |
erdos_1072b : answer(sorry) ↔ ∃ (P : Set ℕ), P ⊆ {p | p.Prime} ∧ P.HasDensity 1 {p | p.Prime} ∧ Tendsto (fun p => (f p / p : ℝ)) (atTop ⊓ principal P) (𝓝 0) := by sorry /-- Erdős, Hardy, and Subbarao [HaSu02], believed that the number of $p \le x$ for which $f(p)=p−1$ is $o(x/\log x)$. [HaSu02] Hardy, G. E. and Subbarao, M. V., _A modified problem of Pillai and some related questions._ Amer. Math. Monthly (2002), 554--559. -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1072.lean | erdos_1072b | /-- Is it true that $f(p)/p \to 0$ for $p \to \infty$ in a density 1 subset of the primes? -/ |
erdos_1072a.variants.littleo : (fun x ↦ (({p | p.Prime ∧ f p = p - 1}.interIcc 0 x).ncard : ℝ)) =o[atTop] (fun x ↦ x / Real.log x) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1072.lean | erdos_1072a.variants.littleo | /--
Erdős, Hardy, and Subbarao [HaSu02], believed that the number of $p \le x$ for which $f(p)=p−1$
is $o(x/\log x)$.
[HaSu02] Hardy, G. E. and Subbarao, M. V., _A modified problem of Pillai and some related questions._
Amer. Math. Monthly (2002), 554--559.
-/ |
A (x : ℕ) : ℝ := {u | u.Composite ∧ ∃ n, n ! + 1 ≡ 0 [MOD u] ∧ u < x}.ncard /-- Is it true that $A(x) \le x^{o(1)}$? -/ @[category research open, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1073.lean | A | /--
Let $A(x)$ count the number of composite $u < x$ such that $n!+1 \equiv 0 (\mod u)$ for some $n$.
-/ |
erdos_1073 : answer(sorry) ↔ ∃ (o : ℕ → ℝ), o =o[atTop] (1 : ℕ → ℝ) ∧ ∀ x, A x ≤ x ^ (o x) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1073.lean | erdos_1073 | /-- Is it true that $A(x) \le x^{o(1)}$? -/ |
Nat.EHSNumbers : Set ℕ := {m | 1 ≤ m ∧ ∃ p, p.Prime ∧ ¬p ≡ 1 [MOD m] ∧ p ∣ m ! + 1} /-- The Pillai primes are those primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$-/ | abbrev | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | Nat.EHSNumbers | /-- The EHS numbers (after Erdős, Hardy, and Subbarao) are those $m\geq 1$ such that there
exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$.-/ |
Nat.PillaiPrimes : Set ℕ := {p | p.Prime ∧ ∃ m, ¬p ≡ 1 [MOD m] ∧ p ∣ m ! + 1} namespace Erdos1074 open Nat /-- Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$. Does $$ \lim\frac{|S\cap[1, x]|}{x} $$ exist? -/ @[category research open, AMS 11] | abbrev | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | Nat.PillaiPrimes | /-- The Pillai primes are those primes $p$ such that there exists an $m$ with
$p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$-/ |
erdos_1074.part8_i_i : answer(sorry) ↔ ∃ c, EHSNumbers.HasDensity c := by sorry /-- Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$. What is $$ \lim\frac{|S\cap[1, x]|}{x}? $$ -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.part8_i_i | /-- Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such
that $m! + 1 \equiv 0\pmod{p}$. Does
$$
\lim\frac{|S\cap[1, x]|}{x}
$$
exist? -/ |
erdos_1074.part_i_ii : EHSNumbers.HasDensity answer(sorry) := by sorry /-- Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$, then does $$ \lim\frac{|P\cap[1, x]|}{\pi(x)} $$ exist? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.part_i_ii | /-- Let $S$ be the set of all $m\geq 1$ such that there exists a prime $p\not\equiv 1\pmod{m}$ such
that $m! + 1 \equiv 0\pmod{p}$. What is
$$
\lim\frac{|S\cap[1, x]|}{x}?
$$ -/ |
erdos_1074.part_ii_i : answer(sorry) ↔ ∃ c, PillaiPrimes.HasDensity c {p | p.Prime} := by sorry /-- Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with $p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$, then what is $$ \lim\frac{|P\cap[1, x]|}{\pi(x)}? $$ -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.part_ii_i | /-- Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with
$p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$, then does
$$
\lim\frac{|P\cap[1, x]|}{\pi(x)}
$$
exist? -/ |
erdos_1074.parts_ii_ii : PillaiPrimes.HasDensity answer(sorry) {p | p.Prime} := by sorry /-- Pillai [Pi30] raised the question of whether there exist any primes in $P$. This was answered by Chowla, who noted that, for example, $14! + 1 \equiv 18! + 1 \equiv 0 \pmod{23}$.-/ @[category test, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.parts_ii_ii | /-- Similarly, if $P$ is the set of all primes $p$ such that there exists an $m$ with
$p\not\equiv 1\pmod{m}$ such that $m! + 1 \equiv 0\pmod{p}$, then what is
$$
\lim\frac{|P\cap[1, x]|}{\pi(x)}?
$$ -/ |
erdos_1074.variants.mem_pillaiPrimes : 23 ∈ PillaiPrimes := by norm_num exact ⟨14, by decide⟩ /-- Erdős, Hardy, and Subbarao proved that $S$ is infinite. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.variants.mem_pillaiPrimes | /-- Pillai [Pi30] raised the question of whether there exist any primes in $P$. This was answered
by Chowla, who noted that, for example, $14! + 1 \equiv 18! + 1 \equiv 0 \pmod{23}$.-/ |
erdos_1074.variants.EHSNumbers_infinite : EHSNumbers.Infinite := by sorry /-- Erdős, Hardy, and Subbarao proved that $P$ is infinite. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.variants.EHSNumbers_infinite | /-- Erdős, Hardy, and Subbarao proved that $S$ is infinite. -/ |
erdos_1074.variants.PillaiPrimes_infinite : PillaiPrimes.Infinite := by sorry /-- The sequence $S$ begins $8, 9, 13, 14, 15, 16, 17, ...$ -/ @[category test, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.variants.PillaiPrimes_infinite | /-- Erdős, Hardy, and Subbarao proved that $P$ is infinite. -/ |
erdos_1074.variants.EHSNumbers_init : nth EHSNumbers '' (Set.Icc 0 6) = {8, 9, 13, 14, 15, 16, 17} := by sorry /-- The sequence $P$ begins $23, 29, 59, 61, 67, 71, ...$ -/ @[category test, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.variants.EHSNumbers_init | /-- The sequence $S$ begins $8, 9, 13, 14, 15, 16, 17, ...$ -/ |
erdos_1074.variants.PillaiPrimes_init : nth PillaiPrimes '' (Set.Icc 0 5) = {23, 29, 59, 61, 67, 71} := by sorry /-- Regarding the first question, Hardy and Subbarao computed all EHS numbers up to $2^{10}$, and write "...if this trend conditions we expect [the limit] to be around 0.5, if it exists." -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.variants.PillaiPrimes_init | /-- The sequence $P$ begins $23, 29, 59, 61, 67, 71, ...$ -/ |
erdos_1074.variants.EHSNumbers_one_half : EHSNumbers.HasDensity (1 / 2) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1074.lean | erdos_1074.variants.EHSNumbers_one_half | /-- Regarding the first question, Hardy and Subbarao computed all EHS numbers up to $2^{10}$, and
write "...if this trend conditions we expect [the limit] to be around 0.5, if it exists." -/ |
erdos_1077 : answer(False) ↔ ∀ ε > (0 : ℝ), ε < 1 → ∀ α > (0 : ℝ), α < 1 → ∀ᶠ D in atTop, ∀ᶠ n in atTop, ∀ G : SimpleGraph (Fin n), G.edgeSet.ncard > (n : ℝ) ^ (1 + α) → ∃ (H : Subgraph G), letI m := H.verts.ncard IsBalanced H.coe D ∧ m > (n : ℝ) ^ (1 - α) ∧ H.edgeSet.ncard > ε * m ^ (1 + α) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1077.lean | erdos_1077 | /--
We call a graph $D$-balanced (or $D$-almost-regular) if the maximum degree is at most $D$ times the
minimum degree.
Let $ε, α > 0$ and $D$ and $n$ be sufficiently large. If $G$ is a graph on $n$ vertices with at
least $n^{1+α}$ edges, then must $G$ contain a $D$-balanced subgraph on $m > n^{1-α}$ vertices with
at least $εm^{1+α}$ edges?
-/ |
erdos_108 : answer(sorry) ↔ ∀ r ≥ 4, ∀ k ≥ (2 : ℕ), ∃ (f : ℕ), ∀ (V : Type u) (G : SimpleGraph V) (_ : Nonempty V) (hchro : f ≤ SimpleGraph.chromaticNumber G), ∃ (H : G.Subgraph), (SimpleGraph.girth H.coe ≥ r) ∧ (SimpleGraph.chromaticNumber H.coe ≥ k) := by sorry -- TODO: Proof for the case r=4 and statement for the infinite case | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/108.lean | erdos_108 | /--
For every r ≥ 4 and k ≥ 2 is there some finite f(k,r) such that every graph of chromatic number ≥ f(k,r)
contains a subgraph of girth ≥ r and chromatic number ≥ k?
-/ |
IsBipartition {V : Type*} (G : SimpleGraph V) (X Y : Set V) : Prop := Disjoint X Y ∧ X ∪ Y = Set.univ ∧ ∀ ⦃u v⦄, G.Adj u v → (u ∈ X ↔ v ∈ Y) /-- Let $G$ be a bipartite graph on $n$ vertices such that one part has $\lfloor n^{2/3}\rfloor$ vertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must contain a $C_6$? -/ @[category research open, AMS 5] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1080.lean | IsBipartition | /-- `IsBipartition G X Y` means that `X` and `Y` form a bipartition of the vertices of `G`. -/ |
erdos_1080 : answer(sorry) ↔ ∃ c > (0 : ℝ), ∀ (V : Type) [Fintype V] [Nonempty V] (G : SimpleGraph V) (X Y : Set V), IsBipartition G X Y → X.ncard = ⌊(Fintype.card V : ℝ) ^ (2/3 : ℝ)⌋₊ → G.edgeSet.ncard ≥ c * Fintype.card V → ∃ (v : V) (walk : G.Walk v v), walk.IsCycle ∧ walk.length = 6 := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1080.lean | erdos_1080 | /--
Let $G$ be a bipartite graph on $n$ vertices such that one part has $\lfloor n^{2/3}\rfloor$
vertices. Is there a constant $c>0$ such that if $G$ has at least $cn$ edges then $G$ must
contain a $C_6$?
-/ |
f (d n : ℕ) : ℕ := ⨆ (s : Finset (ℝ^ d)) (_ : s.card = n) (_ : IsSeparated 1 s.toSet), unitDistNum s /-- It is easy to check that $f_1(n) = n - 1$. -/ @[category research solved, AMS 52] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1084.lean | f | /-- The maximal number of pairs of points which are distance 1 apart that a set of `n` 1-separated
points in `ℝ^d` make. -/ |
erdos_1084_upper_d1 (n : ℕ) : f 1 n = n - 1 := sorry /-- It is easy to check that $f_2(n) < 3n$. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1084.lean | erdos_1084_upper_d1 | /-- It is easy to check that $f_1(n) = n - 1$. -/ |
erdos_1084_easy_upper_d2 (hn : n ≠ 0) : f 2 n < 3 * n := sorry /-- Erdős showed that there is some constant $c > 0$ such that $f_2(n) < 3n - c n^{1/2}$. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1084.lean | erdos_1084_easy_upper_d2 | /-- It is easy to check that $f_2(n) < 3n$. -/ |
erdos_1084_upper_d2 : ∃ c > (0 : ℝ), ∀ n, f 2 n < 3 * n - c * sqrt n := sorry /-- Erdős conjectured that the triangular lattice is best possible in 2D, in particular that $f_2(3n^2 + 3n + 1) < 9n^2 + 6n$. -/ @[category research open, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1084.lean | erdos_1084_upper_d2 | /-- Erdős showed that there is some constant $c > 0$ such that $f_2(n) < 3n - c n^{1/2}$. -/ |
erdos_1084_triangular_optimal_d2 : f 2 (3 * n ^ 2 + 3 * n + 1) = 9 * n ^ 2 + 6 * n := sorry /-- Erdős claims the existence of two constants $c_1, c_2 > 0$ such that $6n - c_1 n^{2/3} ≤ f_3(n) \le 6n - c_2 n^{2/3}$. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1084.lean | erdos_1084_triangular_optimal_d2 | /-- Erdős conjectured that the triangular lattice is best possible in 2D, in particular that
$f_2(3n^2 + 3n + 1) < 9n^2 + 6n$. -/ |
erdos_1084_upper_lower_d3 : ∃ c₁ : ℝ, ∃ c₂ > (0 : ℝ), ∀ᶠ n in atTop, 6 * n - c₁ * n ^ (2 / 3 : ℝ) ≤ f 3 n ∧ f 3 n ≤ 6 * n - c₂ * n ^ (2 / 3 : ℝ) := sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1084.lean | erdos_1084_upper_lower_d3 | /-- Erdős claims the existence of two constants $c_1, c_2 > 0$
such that $6n - c_1 n^{2/3} ≤ f_3(n) \le 6n - c_2 n^{2/3}$. -/ |
f (d n : ℕ) : ℕ := ⨆ (s : Finset (ℝ^ d)) (_ : s.card = n), unitDistNum s /-- Erdős showed $f_2(n) > n^{1+c/\log\log n}$ for some $c > 0$. -/ @[category research solved, AMS 52] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1085.lean | f | /-- The maximal number of pairs of points which are distance 1 apart that a set of `n` points in
`ℝ^d` make. -/ |
erdos_1085_lower_d2 : ∃ c > (0 : ℝ), ∀ᶠ n : ℕ in atTop, (n : ℝ) ^ (1 + c / log (log n)) < f 2 n := sorry /-- Spencer, Szemerédi, and Trotter showed $f_2(n) = O(n^{4/3})$. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1085.lean | erdos_1085_lower_d2 | /-- Erdős showed $f_2(n) > n^{1+c/\log\log n}$ for some $c > 0$. -/ |
erdos_1085_upper_d2 : (fun n ↦ (f 2 n : ℝ)) =O[atTop] (fun n ↦ (n : ℝ) ^ (4/3 : ℝ)) := sorry /-- Erdős showed $f_3(n) = Ω(n^{4/3}\log\log n)$. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1085.lean | erdos_1085_upper_d2 | /-- Spencer, Szemerédi, and Trotter showed $f_2(n) = O(n^{4/3})$. -/ |
erdos_1085_lower_d3 : (fun n : ℕ ↦ (n : ℝ) ^ (4/3 : ℝ) * log (log n)) =O[atTop] (fun n ↦ (f 3 n : ℝ)) := sorry /-- Is the $n^{4/3}\log\log n$ lower bound in 3D also an upper bound?. -/ @[category research open, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1085.lean | erdos_1085_lower_d3 | /-- Erdős showed $f_3(n) = Ω(n^{4/3}\log\log n)$. -/ |
erdos_1085_upper_d3 : answer(sorry) ↔ (fun n ↦ (f 3 n : ℝ)) =O[atTop] (fun n : ℕ ↦ (n : ℝ) ^ (4/3 : ℝ) * log (log n)) := sorry /-- Lenz showed that, for $d \ge 4$, $f_d(n) \ge \frac{p - 1}{2p} n^2 - O(1)$ where $p = \lfloor\frac d2\rfloor$. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1085.lean | erdos_1085_upper_d3 | /-- Is the $n^{4/3}\log\log n$ lower bound in 3D also an upper bound?. -/ |
erdos_1085_lower_d4_lenz (hd : 4 ≤ d) : ∃ C : ℝ, ∀ n : ℕ, ↑(d / 2 - 1) / (2 * ↑(d / 2)) * n ^ 2 - C ≤ f d n := sorry /-- Erdős showed that, for $d \ge 4$, $f_d(n) \le \left(\frac{p - 1}{2p} + o(1)\right) n^2$ where $p = \lfloor\frac d2\rfloor$. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1085.lean | erdos_1085_lower_d4_lenz | /-- Lenz showed that, for $d \ge 4$, $f_d(n) \ge \frac{p - 1}{2p} n^2 - O(1)$ where
$p = \lfloor\frac d2\rfloor$. -/ |
erdos_1085_upper_d4_erdos (hd : 4 ≤ d) : ∃ g : ℕ → ℝ, Tendsto g atTop (𝓝 0) ∧ ∀ n, f d n ≤ (↑(d / 2 - 1) / (2 * ↑(d / 2)) + g n) * n ^ 2 := sorry /-- Erdős and Pach showed that, for $d \ge 5$ odd, there exist constants $c_1(d), c_2(d) > 0$ such that $\frac{p - 1}{2p} n^2 - c_1 n^{4/3} ≤ f_d(n) \le \frac{p - 1}{2p} n^2 + c_2 n^{4/3}$ where $p = \lfloor\frac d2\rfloor$. -/ @[category research solved, AMS 52] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1085.lean | erdos_1085_upper_d4_erdos | /-- Erdős showed that, for $d \ge 4$, $f_d(n) \le \left(\frac{p - 1}{2p} + o(1)\right) n^2$ where
$p = \lfloor\frac d2\rfloor$. -/ |
erdos_1085_upper_lower_d5_odd (hd : 5 ≤ d) (hd_odd : Odd d) : ∃ c₁ > (0 : ℝ), ∃ c₂ : ℝ, ∀ᶠ n in atTop, ↑(d / 2 - 1) / (2 * ↑(d / 2)) * n ^ 2 + c₁ * n ^ (4 / 3 : ℝ) ≤ f d n ∧ f d n ≤ ↑(d / 2 - 1) / ↑(d / 2) * n ^ 2 + c₂ * n ^ (4 / 3 : ℝ) := sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1085.lean | erdos_1085_upper_lower_d5_odd | /-- Erdős and Pach showed that, for $d \ge 5$ odd, there exist constants $c_1(d), c_2(d) > 0$
such that $\frac{p - 1}{2p} n^2 - c_1 n^{4/3} ≤ f_d(n) \le \frac{p - 1}{2p} n^2 + c_2 n^{4/3}$ where
$p = \lfloor\frac d2\rfloor$. -/ |
f (r n : ℕ) : ℕ := sSup {k : ℕ | ∀ G : SimpleGraph (Fin n), (∀ H : Subgraph G, ∃ E : Finset (Sym2 H.verts), E.card ≤ k ∧ chromaticNumber (H.coe.deleteEdges E) ≤ (r + 1 : ℕ∞)) → chromaticNumber G ≤ (r + 1 : ℕ∞)} @[category research open, AMS 5] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1092.lean | f | /--
$f_r(n)$ is maximal such that, if a graph $G$ on $n$ vertices has the property that every
subgraph $H$ on $m$ vertices has chromatic number $\leq r+1$ once we remove $f_r(m)$ edges
from it.
-/ |
f_asymptotic_2 : answer(sorry) ↔ (fun (n : ℕ) => (n : ℝ)) =o[atTop] (fun (n : ℕ) => (f 2 n : ℝ)) := by sorry @[category research open, AMS 5] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1092.lean | f_asymptotic_2 | null |
f_asymptotic_general : answer(sorry) ↔ ∀ r : ℕ, (fun n : ℕ => ((r : ℝ) * n)) =o[atTop] (fun n : ℕ => (f r n : ℝ)) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1092.lean | f_asymptotic_general | null |
deficiency (n k : ℕ) : ℕ := #{i ∈ range k | n - i ∈ smoothNumbers k} /-- Are there infinitely many binomial coefficients with deficiency 1? -/ @[category research open, AMS 5] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1093.lean | deficiency | /--
If defined, the deficiency is the count of $0 \le i < k$ such that $n - i$ is $k$-smooth.
-/ |
erdos_1093.parts.i : answer(sorry) ↔ {x : ℕ × ℕ | let k := x.1; let n := x.2; 2 * k ≤ n ∧ deficiency n k = 1 ∧ ∀ p, p.Prime → (p ∣ choose n k) → k < p}.Infinite := by sorry /-- Are there only finitely many binomial coefficients with deficiency > 1? -/ @[category research open, AMS 5] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1093.lean | erdos_1093.parts.i | /--
Are there infinitely many binomial coefficients with deficiency 1?
-/ |
erdos_1093.parts.ii : {x : ℕ × ℕ | let k := x.1; let n := x.2; 2 * k ≤ n ∧ deficiency n k > 1 ∧ ∀ p, p.Prime → (p ∣ choose n k) → k < p}.Finite := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1093.lean | erdos_1093.parts.ii | /--
Are there only finitely many binomial coefficients with deficiency > 1?
-/ |
erdos_1094 : {(n, k) : ℕ × ℕ | 0 < k ∧ 2 * k ≤ n ∧ (n.choose k).minFac > max (n / k) k}.Finite := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1094.lean | erdos_1094 | /--
**Erdős problem 1094**
There are only finitely many pairs `(n,k)` with `n ≥ 2*k` for which the
least prime factor of the binomial coefficient `Nat.choose n k` exceeds `max (n / k) k`.
-/ |
g (k : ℕ) : ℕ := sSup {m | ∀ n ∈ Set.Ioc k m, ∃ p ≤ k, p.Prime ∧ p ∣ choose n k} /-- The current record is\[g(k) \gg \exp(c(\log k)^2)\]for some $c>0$, due to Konyagin [Ko99b](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300007555). -/ @[category research solved, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1095.lean | g | /--
Let $g(k)>k+1$ be maximal such that
if $n\leq g(k)$ then $\binom{n}{k}$ is divisible by a prime $\leq k$.
Estimate $g(k)$.
-/ |
erdos_1095_lower_solved : ∃ c > 0, (fun k : ℕ ↦ exp (c * log k ^ 2)) =O[atTop] fun k ↦ (g k : ℝ) := by sorry /-- Ecklund, Erdős, and Selfridge conjectured $g(k)\leq \exp(k^{1+o(1)})$ [EES74](https://mathscinet.ams.org/mathscinet/relay-station?mr=1199990) -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1095.lean | erdos_1095_lower_solved | /--
The current record is\[g(k) \gg \exp(c(\log k)^2)\]for some $c>0$, due to Konyagin
[Ko99b](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/S0025579300007555).
-/ |
erdos_1095_upper_conjecture : ∃ f : ℕ → ℝ, Tendsto f atTop (𝓝 0) ∧ ∀ k, g k ≤ exp (k ^ (1 + f k)) := by sorry /-- Erdős, Lacampagne, and Selfridge [ELS93](https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199990-6/S0025-5718-1993-1199990-6.pdf) write 'it is clear to every right-thinking person' that $g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$. -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1095.lean | erdos_1095_upper_conjecture | /--
Ecklund, Erdős, and Selfridge conjectured $g(k)\leq \exp(k^{1+o(1)})$
[EES74](https://mathscinet.ams.org/mathscinet/relay-station?mr=1199990)
-/ |
erdos_1095_lower_conjecture : ∃ c > 0, ∀ k, g k ≥ exp (c * k / log k) := sorry /-- [Sorenson, Sorenson, and Webster](https://mathscinet.ams.org/mathscinet/relay-station?mr=4235124) give heuristic evidence that \[\log g(k) \asymp \frac{k}{\log k}.\] -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1095.lean | erdos_1095_lower_conjecture | /--
Erdős, Lacampagne, and Selfridge [ELS93](https://www.ams.org/journals/mcom/1993-61-203/S0025-5718-1993-1199990-6/S0025-5718-1993-1199990-6.pdf)
write 'it is clear to every right-thinking person' that
$g(k)\geq\exp(c\frac{k}{\log k})$ for some constant $c>0$.
-/ |
erdos_1095_log_equivalent : (fun k ↦ log (g k)) ~[atTop] (fun k ↦ k / log k) := sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1095.lean | erdos_1095_log_equivalent | /--
[Sorenson, Sorenson, and Webster](https://mathscinet.ams.org/mathscinet/relay-station?mr=4235124)
give heuristic evidence that \[\log g(k) \asymp \frac{k}{\log k}.\]
-/ |
CommonDifferencesThreeTermAP (A : Finset ℤ) : Set ℤ := {d : ℤ | d ≠ 0 ∧ ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, b - a = d ∧ c - b = d} /-- The main conjecture: for any finite set of integers $A$ with $|A| = n$, the number of distinct common differences in three-term arithmetic progressions is $O(n^{3/2})$. -/ @[category research open, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1097.lean | CommonDifferencesThreeTermAP | /--
Given a finite set of integers `A` (modelled as a `Finset ℤ`), the set
`CommonDifferencesThreeTermAP A` consists of all integers `d` such that there
is a non-trivial three-term arithmetic progression `a, b, c ∈ A` with
`b - a = d` and `c - b = d`.
-/ |
erdos_1097 : answer(sorry) ↔ ∃ C > (0 : ℝ), ∀ (A : Finset ℤ), (CommonDifferencesThreeTermAP A).ncard ≤ C * (A.card : ℝ) ^ (3 / 2 : ℝ) := by sorry /-- A weaker bound has been proven: there are always at most $n^2$ such values of $d$. -/ @[category undergraduate, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1097.lean | erdos_1097 | /--
The main conjecture: for any finite set of integers $A$ with $|A| = n$, the number of distinct
common differences in three-term arithmetic progressions is $O(n^{3/2})$.
-/ |
erdos_1097.variants.weaker : ∀ A, (CommonDifferencesThreeTermAP A).ncard ≤ A.card ^ 2 := by sorry /-- A trivial lower bound: there exist sets $A$ with $|A| = n$ that contain at least $\Omega(n)$ distinct common differences of three-term arithmetic progressions. -/ @[category undergraduate, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1097.lean | erdos_1097.variants.weaker | /--
A weaker bound has been proven: there are always at most $n^2$ such values of $d$.
-/ |
erdos_1097.variants.lower_bound : ∃ c > (0 : ℝ), ∀ (n : ℕ), ∃ (A : Finset ℤ), A.card = n ∧ c * (n : ℝ) ≤ (CommonDifferencesThreeTermAP A).ncard := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1097.lean | erdos_1097.variants.lower_bound | /--
A trivial lower bound: there exist sets $A$ with $|A| = n$ that contain at least $\Omega(n)$
distinct common differences of three-term arithmetic progressions.
-/ |
erdos_11 (n : ℕ) (hn : Odd n) (hn' : 1 < n) : ∃ k l : ℕ, Squarefree k ∧ n = k + 2 ^ l := by sorry /-- Erdős often asked this under the weaker assumption that $n > 1$ is not divisible by 4. -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/11.lean | erdos_11 | /--
Is every odd $n > 1$ the sum of a squarefree number and a power of 2?
-/ |
erdos_11.variants.not_four_dvd (n : ℕ) (hn : ¬ 4 ∣ n) (hn' : 1 < n) : ∃ k l : ℕ , Squarefree k ∧ n = k + 2^l := by sorry /-- Is every odd $n > 1$ the sum of a squarefree number and two powers of 2? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/11.lean | erdos_11.variants.not_four_dvd | /--
Erdős often asked this under the weaker assumption that $n > 1$
is not divisible by 4.
-/ |
erdos_11.variants.two_pow_two (n : ℕ) (hn : Odd n) (hn' : 1 < n) : ∃ k l m : ℕ , Squarefree k ∧ n = k + 2^l + 2^m := by sorry /-- Every odd $1 < n < 10^7$ is the sum of a squarefree number and a power of 2. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/11.lean | erdos_11.variants.two_pow_two | /--
Is every odd $n > 1$ the sum of a squarefree number and two powers of 2?
-/ |
erdos_11.variants.finite_bound1 (n : ℕ) (hn : Odd n) (h : n < 10^7) (hn' : 1 < n) : ∃ k l : ℕ , Squarefree k ∧ n = k + 2^l := by sorry /-- Every odd $1 < n < 2^50$ is the sum of a squarefree number and a power of 2. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/11.lean | erdos_11.variants.finite_bound1 | /--
Every odd $1 < n < 10^7$ is the sum of a squarefree number and a power of 2.
-/ |
erdos_11.variants.finite_bound2 (n : ℕ) (hn : Odd n) (h : n < 2^50) (hn' : 1 < n) : ∃ k l : ℕ , Squarefree k ∧ n = k + 2^l := by sorry /-- Suppose that every odd $n$ is the sum of a squarefree number and a power of 2. Then the set of primes $p$ such that $2 ^ p ≡ 2 \mod p ^ 2$ is infinite. This is Theorem 1 in [GrSo98]. [GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/11.lean | erdos_11.variants.finite_bound2 | /--
Every odd $1 < n < 2^50$ is the sum of a squarefree number and a power of 2.
-/ |
erdos_11.variants.granville_soundararajan (H : type_of% erdos_11) : {p : ℕ | p.Prime ∧ 2 ^ p ≡ 2 [MOD p ^ 2]}.Infinite := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/11.lean | erdos_11.variants.granville_soundararajan | /--
Suppose that every odd $n$ is the sum of a squarefree number and a power of 2. Then the set of primes
$p$ such that $2 ^ p ≡ 2 \mod p ^ 2$ is infinite. This is Theorem 1 in [GrSo98].
[GrSo98] Granville, A. and Soundararajan, K., A Binary Additive Problem of Erdős and the Order of $2$ mod $p^2$. The Ramanujan Journal (1998), 283-298.
-/ |
ASet (u : ℕ → ℕ) : Set ℕ := { a | ∀ i, ¬ u i ∣ a } /-- The sequence of integers A_u which are not divisible by any u_i arranged in a monotonic sequence. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1101.lean | ASet | /-- The set of integers not divisible by any u_i. -/ |
A (u : ℕ → ℕ) (n : ℕ) : ℕ := Nat.nth (fun a => a ∈ ASet u) n /-- t_x such that u_0 ... u_{t_x-1} ≤ x < u_0 ... u_{t_x}. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1101.lean | A | /-- The sequence of integers A_u which are not divisible by any u_i
arranged in a monotonic sequence. -/ |
t (u : ℕ → ℕ) (x : ℕ) : ℕ := sSup { k | ∏ i ∈ Finset.range k, u i ≤ x } /-- A sequence is "good" if 1. it is strictly monotone 2. it is pairwise coprime 3. the sum of reciprocals converges 4. the gap between consecutive elements in A(u) is bounded relative to t_x. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1101.lean | t | /-- t_x such that u_0 ... u_{t_x-1} ≤ x < u_0 ... u_{t_x}.
-/ |
IsGood (u : ℕ → ℕ) : Prop := StrictMono u ∧ (∀ i j, i ≠ j → Coprime (u i) (u j)) ∧ Summable (fun n => 1 / (u n : ℝ)) ∧ ∀ ε > 0, ∀ᶠ x in atTop, ∀ k, A u k < x → (A u (k + 1) : ℝ) - A u k < (1 + ε) * (t u x : ℝ) * (∏' i : ℕ, (1 - 1 / (u i : ℝ)))⁻¹ /-- 1. There is NO good sequence with polynomial growth. -/ @[category research open, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1101.lean | IsGood | /-- A sequence is "good" if
1. it is strictly monotone
2. it is pairwise coprime
3. the sum of reciprocals converges
4. the gap between consecutive elements in A(u) is bounded relative to t_x. -/ |
erdos_1101.polynomial : ¬ ∃ u, IsGood u ∧ ∃ k : ℕ, (fun n => (u n : ℝ)) =O[atTop] (fun n => (n : ℝ) ^ k) := sorry /-- 2. There is a good sequence with sub-exponential growth. -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1101.lean | erdos_1101.polynomial | /-- 1. There is NO good sequence with polynomial growth. -/ |
erdos_1101.subexponential : ∃ u, IsGood u ∧ (fun n => Real.log (u n : ℝ)) =o[atTop] (fun n => (n : ℝ)) := sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1101.lean | erdos_1101.subexponential | /-- 2. There is a good sequence with sub-exponential growth. -/ |
HasPropertyP (A : Set ℕ) : Prop := ∀ n ≥ 1, {a ∈ A | Squarefree (n + a)}.Finite /-- Property Q : A set $A ⊆ ℕ $ has property Q, if the set $\{n ∈ ℕ | ∀ a ∈ A, n > a\text{ implies }n + a\text{ is squarefree}\}$ is infinite. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1102.lean | HasPropertyP | /--
Property P : A set $A ⊆ ℕ $ has property P, if for all $n ≥ 1$ the set
$ \{a ∈ A | n + a\text{ is squarefree}\}$ is finite.
-/ |
HasPropertyQ (A : Set ℕ) : Prop := {n : ℕ | ∀ a ∈ A, a < n → Squarefree (n + a)}.Infinite /-- If `A = {a₁ < a₂ < …}` has property P, then `A` has natural density `0`. Equivalently, `(a_j / j) → ∞` as `j → ∞`. -/ @[category research solved, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1102.lean | HasPropertyQ | /--
Property Q : A set $A ⊆ ℕ $ has property Q, if the set
$\{n ∈ ℕ | ∀ a ∈ A, n > a\text{ implies }n + a\text{ is squarefree}\}$ is infinite.
-/ |
erdos_1102.density_zero_of_P (A : ℕ → ℕ) (h_inc : StrictMono A) (hP : HasPropertyP (range A)) : Tendsto (fun j => (A j / j : ℝ)) atTop atTop := by sorry /-- Conversely, for any function `f : ℕ → ℕ` that goes to infinity, there exists a strictly increasing sequence `A = {a₁ < a₂ < …}` with property P such that `(a_j / j) ≤ f(j)` for all `j`. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1102.lean | erdos_1102.density_zero_of_P | /--
If `A = {a₁ < a₂ < …}` has property P,
then `A` has natural density `0`.
Equivalently, `(a_j / j) → ∞` as `j → ∞`.
-/ |
erdos_1102.exists_sequence_with_P (f : ℕ → ℕ) (h_inf : Tendsto f atTop atTop) (h_pos : ∀ n, f n ≠ 0) : ∃ A : ℕ → ℕ, StrictMono A ∧ HasPropertyP (range A) ∧ ∀ j : ℕ, (A j : ℝ) / j ≤ f j := by sorry /-- Every sequence with property Q has upper density at most `6 / π^2`. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1102.lean | erdos_1102.exists_sequence_with_P | /--
Conversely, for any function `f : ℕ → ℕ` that goes to infinity,
there exists a strictly increasing sequence `A = {a₁ < a₂ < …}`
with property P such that `(a_j / j) ≤ f(j)` for all `j`.
-/ |
erdos_1102.upper_density_Q (A : ℕ → ℕ) (h_inc : StrictMono A) (hQ : HasPropertyQ (range A)) : limsup (fun j : ℕ ↦ j / A j) atTop ≤ 6 / Real.pi^2 := by sorry /-- There exists an infinite sequence $A = {a₁ < a₂ < …} ⊂ \mathsf{SF}$ where $\mathsf{SF} := \mathbb{N} \setminus \bigcup_{p} p^{2}\mathbb{N}$, i.e. the set of squarefree numbers. The set `A` has property `Q` and natural density `6 / π^2`. Equivalently, `(j / a_j) → 6/π^2` as `j → ∞`. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1102.lean | erdos_1102.upper_density_Q | /--
Every sequence with property Q has upper density at most `6 / π^2`.
-/ |
erdos_1102.lower_density_Q_exists : ∃ A : ℕ → ℕ, StrictMono A ∧ (∀ j, Squarefree (A j)) ∧ HasPropertyQ (range A) ∧ Tendsto (fun j : ℕ ↦ (j / A j : ℝ)) atTop (𝓝 (6 / Real.pi^2)) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1102.lean | erdos_1102.lower_density_Q_exists | /--
There exists an infinite sequence $A = {a₁ < a₂ < …} ⊂ \mathsf{SF}$ where
$\mathsf{SF} := \mathbb{N} \setminus \bigcup_{p} p^{2}\mathbb{N}$, i.e. the set of
squarefree numbers. The set `A` has property `Q` and natural density `6 / π^2`.
Equivalently, `(j / a_j) → 6/π^2` as `j → ∞`.
-/ |
triangleFreeMaxChromatic (n : ℕ) : ℕ := sSup {χ | ∃ G : SimpleGraph (Fin n), G.CliqueFree 3 ∧ G.chromaticNumber = χ} /-- Lower bound (Hefty–Horn–King–Pfender 2025). There exists a constant $c_1 \in (0,1]$ such that, for sufficiently large $n$, $$ c_1 \sqrt{\frac{n}{\log n}} \le f(n), $$ where $f(n)$ denotes the maximum chromatic number of a triangle-free graph on $n$ vertices, formalized as `triangleFreeMaxChromatic n`. -/ @[category research solved, AMS 5] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1104.lean | triangleFreeMaxChromatic | /-- Maximum chromatic number of a triangle-free graph on `n` vertices. -/ |
erdos_1104_lower : ∃ c₁ : ℝ, 0 < c₁ ∧ c₁ ≤ 1 ∧ (∀ᶠ n : ℕ in atTop, c₁ * Real.sqrt (n : ℝ) / Real.sqrt (Real.log (n : ℝ)) ≤ (triangleFreeMaxChromatic n : ℝ)) := by sorry /-- Upper bound (Davies–Illingworth 2022). There exists a constant $c_2 \ge 2$ such that, for sufficiently large $n$, $$ f(n) \le c_2 \sqrt{\frac{n}{\log n}}, $$ where $f(n)$ denotes the maximum chromatic number of a triangle-free graph on $n$ vertices, formalized as `triangleFreeMaxChromatic n`. -/ @[category research solved, AMS 5] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1104.lean | erdos_1104_lower | /--
Lower bound (Hefty–Horn–King–Pfender 2025).
There exists a constant $c_1 \in (0,1]$ such that, for sufficiently large $n$,
$$
c_1 \sqrt{\frac{n}{\log n}} \le f(n),
$$
where $f(n)$ denotes the maximum chromatic number of a triangle-free graph on
$n$ vertices, formalized as `triangleFreeMaxChromatic n`.
-/ |
erdos_1104_upper : ∃ c₂ : ℝ, 2 ≤ c₂ ∧ (∀ᶠ n : ℕ in atTop, (triangleFreeMaxChromatic n : ℝ) ≤ c₂ * Real.sqrt (n : ℝ) / Real.sqrt (Real.log (n : ℝ))) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1104.lean | erdos_1104_upper | /--
Upper bound (Davies–Illingworth 2022).
There exists a constant $c_2 \ge 2$ such that, for sufficiently large $n$,
$$
f(n) \le c_2 \sqrt{\frac{n}{\log n}},
$$
where $f(n)$ denotes the maximum chromatic number of a triangle-free graph on
$n$ vertices, formalized as `triangleFreeMaxChromatic n`.
-/ |
p : ℕ → ℕ := fun n => Fintype.card (Nat.Partition n) /-- Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity. -/ @[category research open, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1106.lean | p | /-- The partition function p(n) is the number of ways to write n as a sum of positive
integers (where the order of the summands does not matter). -/ |
erdos_1106 : Tendsto (fun n => #(∏ i ∈ Icc 1 n, p i).primeFactors) atTop atTop := sorry /-- Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of $∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$. -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1106.lean | erdos_1106 | /--
Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of
$∏_{i= 1} ^ {n} p(n)$, then $F(n)$ tends to infinity when $n$ tends to infinity.
-/ |
erdos_1106_k2 : ∀ᶠ n in atTop, #(∏ i ∈ Icc 1 n, p i).primeFactors > n := sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1106.lean | erdos_1106_k2 | /--
Let $p(n)$ be the partition number of $n$ and $F(n)$ be the number of distinct prime factors of
$∏_{i= 1} ^ {n} p(n)$, $F(n)>n$ for sufficiently large $n$.
-/ |
SumOfRPowerful (r n : ℕ) : Prop := ∃ s : List ℕ, s.length ≤ r + 1 ∧ (∀ x ∈ s, Nat.Full r x) ∧ s.sum = n /-- Let $r \ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers? -/ @[category research open, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1107.lean | SumOfRPowerful | /--
Helper Property: $n$ is the sum of at most $r+1$ numbers, each of which is $r$-full.
-/ |
erdos_1107 : ∀ r ≥ 2, ∀ᶠ n in atTop, SumOfRPowerful r n := by sorry /-- Heath-Brown [He88] proved every large integer the sum of at most three $2$-powerful numbers. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1107.lean | erdos_1107 | /--
Let $r \ge 2$. Is every large integer the sum of at most $r + 1$ many $r$-powerful numbers?
-/ |
erdos_1107.variants.two : ∀ᶠ n in atTop, SumOfRPowerful 2 n := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1107.lean | erdos_1107.variants.two | /--
Heath-Brown [He88] proved every large integer the sum of at most three $2$-powerful numbers.
-/ |
FactorialSums : Set ℕ := {m : ℕ | ∃ S : Finset ℕ, m = ∑ n ∈ S, n.factorial} /-- A number is powerful if each prime factor appears with exponent at least 2. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1108.lean | FactorialSums | /--
The set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite
sums of distinct factorials.
-/ |
IsPowerful (n : ℕ) : Prop := ∀ p : ℕ, p.Prime → p ∣ n → p ^ 2 ∣ n /-- For each $k \geq 2$, does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers? -/ @[category research open, AMS 11] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1108.lean | IsPowerful | /--
A number is powerful if each prime factor appears with exponent at least 2.
-/ |
erdos_1108.k_th_powers : answer(sorry) ↔ ∀ k ≥ 2, Set.Finite { a | a ∈ FactorialSums ∧ ∃ m : ℕ, m ^ k = a } := by sorry /-- Does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many powerful numbers? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1108.lean | erdos_1108.k_th_powers | /--
For each $k \geq 2$, does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many $k$-th powers?
-/ |
erdos_1108.powerful_numbers : answer(sorry) ↔ {a ∈ FactorialSums | IsPowerful a}.Finite := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/1108.lean | erdos_1108.powerful_numbers | /--
Does the set $A = \left\{ \sum_{n\in S}n! : S\subset \mathbb{N}\text{ finite}\right\}$ of all finite sums of distinct factorials contain only finitely many powerful numbers?
-/ |
p (z : ℕ → ℂ) (n : ℕ) : ℂ → ℂ := fun w => ∏ i ∈ range n, (w - z i) /-- Let $M_n = \max_{|z| = 1} |p_n(z)|$. -/ | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/119.lean | p | /-- Let $z_i$ be an infinite sequence of complex numbers such that $|z_i| = 1$ for all $i \geq 1$.
For $n \geq 1$ let $p_n(z) = \prod_{i \leq n} (z - z_i)$. -/ |
M (z : ℕ → ℂ) (n : ℕ) : ℝ := sSup { (‖p z n w‖) | (w : ℂ) (_ : ‖w‖ = 1) } /-- Question 1: Is it true that $\limsup M_n = \infty$? Wagner [Wa80] proved that there is some $c > 0$ with $M_n > (\log n)^c$ infintely often. [Wa80] Wagner, Gerold, On a problem of {E}rdős in {D}iophantine approximation. Bull. London Math. Soc. (1980), 81--88. -/ @[category research solved, AMS 30] | def | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/119.lean | M | /-- Let $M_n = \max_{|z| = 1} |p_n(z)|$. -/ |
erdos_119_1 : answer(True) ↔ ∀ (z : ℕ → ℂ) (hz : ∀ i : ℕ, ‖z i‖ = 1), atTop.limsup (fun n => (M z n : EReal)) = ⊤ := by sorry /-- Question 2: Is it true that there exists $c > 0$ such that for infinitely many $n$ we have $M_n > n^c$? Beck [Be91] proved that there exists some $c > 0$ such that $\max_{n \leq N} M_n > N^c$. [Be91] Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erdős. Annals of Math. (1991), 609-651. -/ @[category research solved, AMS 30] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/119.lean | erdos_119_1 | /-- Question 1:
Is it true that $\limsup M_n = \infty$?
Wagner [Wa80] proved that there is some $c > 0$ with $M_n > (\log n)^c$ infintely often.
[Wa80] Wagner, Gerold, On a problem of {E}rdős in {D}iophantine approximation. Bull. London Math. Soc. (1980), 81--88.
-/ |
erdos_119_2 : answer(True) ↔ ∀ (z : ℕ → ℂ) (hz : ∀ i : ℕ, ‖z i‖ = 1), ∃ (c : ℝ) (hc : c > 0), Infinite {n : ℕ | M z n > n ^ c} := by sorry /-- Question 3: Is it true that there exists $c > 0$ such that, for all large $n$, $\sum_{k \leq n} M_k > n^{1 + c}$? -/ @[category research open, AMS 30] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/119.lean | erdos_119_2 | /-- Question 2:
Is it true that there exists $c > 0$ such that for infinitely many $n$ we have $M_n > n^c$?
Beck [Be91] proved that there exists some $c > 0$ such that $\max_{n \leq N} M_n > N^c$.
[Be91] Beck, J., The modulus of polynomials with zeros on the unit circle: A problem of Erdős. Annals of Math. (1991), 609-651.
-/ |
erdos_119_3 : answer(sorry) ↔ ∀ (z : ℕ → ℂ) (hz : ∀ i : ℕ, ‖z i‖ = 1), ∃ (c : ℝ) (hc : c > 0), ∀ᶠ n in atTop, ∑ k ∈ range n, M z k > n ^ (1 + c) := by sorry | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/119.lean | erdos_119_3 | /-- Question 3:
Is it true that there exists $c > 0$ such that, for all large $n$, $\sum_{k \leq n} M_k > n^{1 + c}$?
-/ |
IsGood (A : Set ℕ) : Prop := A.Infinite ∧ ∀ᵉ (a ∈ A) (b ∈ A) (c ∈ A), a ∣ b + c → a < b → a < c → b = c /-- The set of $p ^ 2$ where $p \cong 3 \mod 4$ is prime is an example of a good set. -/ @[category undergraduate, AMS 11] | abbrev | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/12.lean | IsGood | /--
A set `A` is "good" if it is infinite and there are no distinct `a,b,c` in `A`
such that `a ∣ (b+c)` and `b > a`, `c > a`.
-/ |
isGood_example : IsGood {p ^ 2 | (p : ℕ) (_ : p ≡ 3 [MOD 4]) (_ : p.Prime)} := by sorry open Erdos12 /-- Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Is there such an $A$ with $\liminf \frac{|A \cap \{1, \dotsc, N\}|}{N^{1/2}} > 0$ ? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/12.lean | isGood_example | /-- The set of $p ^ 2$ where $p \cong 3 \mod 4$ is prime is an example of a good set. -/ |
erdos_12.parts.i : answer(sorry) ↔ ∃ (A : Set ℕ), IsGood A ∧ (0 : ℝ) < Filter.atTop.liminf (fun N => (A.interIcc 1 N).ncard / (N : ℝ).sqrt) := by sorry /-- Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Does there exist some absolute constant $c > 0$ such that there are always infinitely many $N$ with $|A \cap \{1, \dotsc, N\}| < N^{1−c}$? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/12.lean | erdos_12.parts.i | /--
Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$
such that $a \mid (b+c)$ and $b,c > a$. Is there such an $A$ with
$\liminf \frac{|A \cap \{1, \dotsc, N\}|}{N^{1/2}} > 0$ ?
-/ |
erdos_12.parts.ii : answer(sorry) ↔ ∃ c > (0 : ℝ), ∀ (A : Set ℕ), IsGood A → {N : ℕ| (A.interIcc 1 N).ncard < (N : ℝ) ^ (1 - c)}.Infinite := by sorry /-- Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$. Is it true that $∑_{n \in A} \frac{1}{n} < \infty$? -/ @[category research open, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/12.lean | erdos_12.parts.ii | /--
Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$
such that $a \mid (b+c)$ and $b,c > a$. Does there exist some absolute constant $c > 0$
such that there are always infinitely many $N$
with $|A \cap \{1, \dotsc, N\}| < N^{1−c}$?
-/ |
erdos_12.parts.iii : answer(sorry) ↔ ∀ (A : Set ℕ), IsGood A → Summable (fun (n : A) ↦ (1 / n : ℝ)) := by sorry /-- Erdős and Sárközy proved that such an $A$ must have density 0. [ErSa70] Erd\H os, P. and Sárk\"ozi, A., On the divisibility properties of sequences of integers. Proc. London Math. Soc. (3) (1970), 97-101 -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/12.lean | erdos_12.parts.iii | /--
Let $A$ be an infinite set such that there are no distinct $a,b,c \in A$
such that $a \mid (b+c)$ and $b,c > a$. Is it true that $∑_{n \in A} \frac{1}{n} < \infty$?
-/ |
erdos_12.variants.erdos_sarkozy_density_0 (A : Set ℕ) (hA : IsGood A) : A.HasDensity 0 := by sorry /-- Given any function $f(x)\to \infty$ as $x\to \infty$ there exists a set $A$ with the property that there are no distinct $a,b,c \in A$ such that $a \mid (b+c)$ and $b,c > a$, such that there are infinitely many $N$ such that \[\lvert A\cap\{1,\ldots,N\}\rvert > \frac{N}{f(N)}. -/ @[category research solved, AMS 11] | theorem | FormalConjectures | [
"import FormalConjectures.Util.ProblemImports"
] | FormalConjectures/ErdosProblems/12.lean | erdos_12.variants.erdos_sarkozy_density_0 | /--
Erdős and Sárközy proved that such an $A$ must have density 0.
[ErSa70] Erd\H os, P. and Sárk\"ozi, A., On the divisibility properties of sequences of integers.
Proc. London Math. Soc. (3) (1970), 97-101
-/ |
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