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258 problems · 3/6Number SourcesGraph view

declared disproved (Lean) · formalizederdos:429

False ↔ ∃ f, Filter.Tendsto f Filter.atTop Filter.atTop ∧ ∀ (A : Set ℕ), A.Infinite → (∀ (N : ℕ), (A ∩ Set.Icc 1 N).ncard ≤ f N) → (∀ (p : ℕ), Nat.Prime p → ∃ b, ∀ a ∈ A, ↑a ≠ b) → ∃ n, ∀ a ∈ A, Nat.Prime (n + a)

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:434

∀ (n k : ℕ), 1 ≤ n → 2 ≤ k → k ≤ n → IsGreatest {x | ∃ S, ∃ (_ : S ⊆ Finset.Icc 1 n) (_ : S.card = k) (_ : S.gcd id = 1), Erdos434.Nat.NcardUnrepresentable ↑S = x} (Erdos434.Nat.NcardUnrepresentable (Set.Icc (n - k + 1) n))

number theorypossible5 sources

declared proved (Lean) · formalizederdos:435

∀ (n : ℕ), n ≠ 0 → (∀ (p k : ℕ), Nat.Prime p → n ≠ p ^ k) → IsGreatest {m | ¬∃ c, m = ∑ i ∈ Finset.Ico 1 n, ↑(c i) * ↑(n.choose i)} (∑ p ∈ n.primeFactors, (∑ d ∈ Finset.Icc 1 (n.factorization p), ↑(n.choose (p ^ d))) * (↑p - 1) - ↑n)

number theoryA3894795 sources

declared proved (Lean) · formalizederdos:443

True ↔ ∀ (s : ℕ), ∃ m, ∃ n < m, s ≤ (Erdos443.A n ∩ Erdos443.A m).card

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:447

True ↔ (fun n => ↑(Erdos447.maxUnionFree n)) =o[Filter.atTop] fun n => 2 ^ n

combinatoricspossible5 sources

declared disproved (Lean) · formalizederdos:453

False ↔ Erdos453.EventuallyHasPrimeWitness

number theoryN/A5 sources

declared open · formalizederdos:456

True ↔ Filter.Tendsto (fun N => ↑(Nat.count {n | Erdos456.m n < Erdos456.p n} N) / ↑N) Filter.atTop (nhds 1)

number theoryA034694 · A061026 · possible3 sources

declared proved (Lean) · formalizederdos:457

True ↔ ∃ ε > 0, {n | ∀ (p : ℕ), ↑p ≤ (2 + ε) * Real.log ↑n → Nat.Prime p → p ∣ ∏ i ∈ Finset.Icc 1 ⌊Real.log ↑n⌋₊, (n + i)}.Infinite

number theoryA3916686 sources

declared solved (Lean) · formalizederdos:459

∀ {u : ℕ}, 2 ≤ u → u + 2 ≤ Erdos459.f u ∧ Erdos459.f u ≤ u ^ 2

number theoryprimesA2892805 sources

declared proved (Lean) · formalizederdos:464

True ↔ ∀ (n : ℕ → ℕ), StrictMono n → (∀ (k : ℕ), 0 < n k) → IsLacunary n → ∃ θ, Irrational θ ∧ ¬Dense (Set.range fun k => ↑(θ * ↑(n k)))

number theoryN/A4 sources

declared proved (Lean) · formalizederdos:476

True ↔ ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (A : Finset (ZMod p)), A.restrictedSumset.card ≥ min (2 * A.card - 3) p

number theoryadditive combinatoricsN/A5 sources

declared open · formalizederdos:477

Does there exist an integer polynomial

f

of degree at least two and a set

A \subseteq \mathbb{Z}

such that every integer has a unique representation

n = a + f(k)

? A manuscript claims the thirteenth powers admit a tiling complement.

number theoryN/A4 sources

declared proved · formalizederdos:480

True ↔ ∀ (x : ℕ → ℝ), (∀ (n : ℕ), x n ∈ Set.Icc 0 1) → ⨅ n, Filter.liminf (fun m => ↑↑n * |x (m + ↑n) - x m|) Filter.atTop ≤ 1 / √5

number theoryN/A3 sources

declared proved (Lean) · formalizederdos:481

∀ {r : ℕ} (a b : Fin r → ℕ), (∀ (i : Fin r), 0 < a i) → 1 < ∑ i, 1 / ↑(a i) → ∃ k, ¬((Erdos481.T a b)^[k] [1]).Nodup

number theoryN/A5 sources

declared proved (Lean) · formalizederdos:484

∃ c, 0 < c ∧ ∀ (k : ℕ), 0 < k → ∃ N₀, ∀ (N : ℕ), N₀ ≤ N → ∀ (f : ℕ → Fin k), c * ↑N ≤ ↑{n ∈ Finset.Icc 1 N | ∃ a ∈ Finset.Icc 1 N, ∃ b ∈ Finset.Icc 1 N, a ≠ b ∧ f a = f b ∧ a + b = n}.card

number theoryadditive combinatoricsramsey theoryN/A5 sources

declared proved (Lean) · formalizederdos:487

True ↔ ∀ (A : Set ℕ), A.HasPosDensity → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, a ≠ b ∧ b ≠ c ∧ a ≠ c ∧ a.lcm b = c

number theoryN/A5 sources

declared falsifiable · formalizederdos:488

True ↔ ∀ (A : Finset ℕ), A.Nonempty → 0 ∉ A → 1 ∉ A → ∀ (n m : ℕ), m > n → A.max ≤ ↑n → ↑{x ∈ Finset.Icc 1 m | x ∈ {n | n ≥ 1 ∧ ∃ a ∈ A, a ∣ n}}.card / ↑m < 2 * ↑{x ∈ Finset.Icc 1 n | x ∈ {n | n ≥ 1 ∧ ∃ a ∈ A, a ∣ n}}.card / ↑n

number theoryN/A3 sources

declared proved (Lean) · formalizederdos:493

True ↔ ∃ k N, ∀ (n : ℤ), N ≤ n → ∃ a, (∀ (i : Fin k), 2 ≤ a i) ∧ ∏ i, a i - ∑ i, a i = n

number theoryN/A5 sources

declared solved (Lean) · formalizederdos:497

∃ o, ∃ (_ : o =o[Filter.atTop] 1), ∀ (n : ℕ), ↑(DedekindNumber.M' n) = 2 ^ ((1 + o n) * ↑(n.choose (n / 2)))

combinatoricsA0003725 sources

declared proved (Lean) · formalizederdos:498

True ↔ ∀ (n : ℕ) (z : Fin n → ℂ), (∀ (i : Fin n), 1 ≤ ‖z i‖) → ∀ (c : ℂ), {ε | (∀ (i : Fin n), ε i = -1 ∨ ε i = 1) ∧ ∑ i, ↑(ε i) * z i ∈ Metric.ball c 1}.ncard ≤ n.choose (n / 2)

combinatoricsanalysisN/A5 sources

declared proved (Lean) · formalizederdos:499

True ↔ ∀ (n : ℕ), ∀ M ∈ doublyStochastic ℝ (Fin n), ∃ σ, ↑n ^ (-↑n) ≤ ∏ i, M i (σ i)

combinatoricsN/A5 sources

declared open · formalizederdos:501

For every

x \in \mathbb{R}

let

A_x \subset \mathbb{R}

be a bounded set of Lebesgue outer measure

< 1

. Must there be an infinite independent set, that is an infinite

X \subseteq \mathbb{R}

with

x \notin A_y

for all distinct

x, y \in X

? Erdős and Hajnal proved that arbitrarily large finite independent sets exist. Hechler showed in 1972 that the answer is no under the continuum hypothesis, so any positive answer had to come from a model where CH fails, and Sungchul Lee later derived one from a real-valued measurable cardinal. The answer is that neither side is provable. Dropping Lee's large cardinal by transferring his argument to the extension of a model of CH by random reals gives a model where the answer is yes; Hechler's construction gives one where it is no. The question is independent of ZFC.

combinatoricsset theoryN/A4 sources

declared solved (Lean) · formalizederdos:502

∀ (n : ℕ) (A : Set (EuclideanSpace ℝ (Fin n))), A.Finite → {d | ∃ x ∈ A, ∃ y ∈ A, x ≠ y ∧ dist x y = d}.ncard = 2 → A.ncard ≤ (n + 2).choose 2

geometrydistancesA0276275 sources

declared open · formalizederdos:503

∀ (n : ℕ), IsGreatest {x | ∃ A, ∃ (_ : A.IsIsosceles), A.ncard = x} sorry

geometrydistancesA1757693 sources

declared disproved (Lean) · formalizederdos:505

∃ n S, Bornology.IsBounded S ∧ 0 < Metric.diam S ∧ ∀ (F : Fin (n + 1) → Set (EuclideanSpace ℝ (Fin n))), S ⊆ ⋃ i, F i → ∃ i, Metric.diam S ≤ Metric.diam (F i)

geometrypossible5 sources

declared open · formalizederdos:507

Asymptotics.IsEquivalent Filter.atTop Erdos507.α sorry

geometryN/A3 sources

declared proved (Lean) · formalizederdos:512

True ↔ ∃ c > 0, ∀ (N : ℕ) (A : Finset ℤ), A.card = N → c * Real.log ↑N ≤ ∫ (θ : ℝ) in 0..1, ‖∑ n ∈ A, additiveChar (↑n * θ)‖

analysisN/A4 sources

declared open · formalizederdos:513

sorry = ⨆ f, Filter.liminf (fun r => Erdos513.ratio r ↑f) Filter.atTop

analysisN/A3 sources

declared proved (Lean) · formalizederdos:519

True ↔ ∃ c, 0 < c ∧ ∀ (n : ℕ) (hn : 0 < n) (z : Fin n → ℂ), z ⟨0, hn⟩ = 1 → ∃ k, c < ‖Erdos519.powerSum z (↑k + 1)‖

analysisN/A5 sources

declared open · formalizederdos:521

False ↔ Erdos521.Claim

analysispolynomialsprobabilityN/A4 sources

declared open · formalizederdos:535

∀ r ≥ 3, ∃ c > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ↑(Erdos535.f r N) ≤ ↑N ^ (c / Real.log (Real.log ↑N))

number theorypossible3 sources

declared open · formalizederdos:536

True ↔ ∀ ε > 0, ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, ε * ↑N ≤ ↑A.card → ∃ a ∈ A, ∃ b ∈ A, ∃ c ∈ A, {a, b, c}.card = 3 ∧ a.lcm b = b.lcm c ∧ b.lcm c = a.lcm c

number theorypossible3 sources

declared disproved (Lean) · formalizederdos:537

False ↔ ∀ (ε : ℝ), 0 < ε → ∀ᶠ (N : ℕ) in Filter.atTop, ∀ A ⊆ Finset.Icc 1 N, ↑A.card ≥ ε * ↑N → ∃ a₁ ∈ A, ∃ a₂ ∈ A, ∃ a₃ ∈ A, ∃ p₁ p₂ p₃, Nat.Prime p₁ ∧ Nat.Prime p₂ ∧ Nat.Prime p₃ ∧ p₁ ≠ p₂ ∧ p₁ ≠ p₃ ∧ p₂ ≠ p₃ ∧ a₁ * p₁ = a₂ * p₂ ∧ a₂ * p₂ = a₃ * p₃

number theorypossible5 sources

declared open · formalizederdos:539

For

|A| = n

, how small can the cofactor set

Q(A) = \{a / \gcd(a,b) : a, b \in A\}

be? The answer is

h(n) = n^{1/2 + o(1)}

: a new upper bound

h(n) \le n^{1/2} \exp(O(\sqrt{\log n}))

matches the classical lower bound.

number theorypossible5 sources

declared proved (Lean) · formalizederdos:540

True ↔ ∃ C, 0 < C ∧ ∀ (N : ℕ), 0 < N → ∀ (A : Finset (ZMod N)), C * √↑N ≤ ↑A.card → Erdos540.HasZeroSubsetSum A

number theoryA0344635 sources

declared proved (Lean) · formalizederdos:541

True ↔ ∀ (p : ℕ), Fact (Nat.Prime p) → ∀ (a : Fin p → ZMod p), (∃ r, ∀ (S : Finset (Fin p)), S ≠ ∅ → ∑ i ∈ S, a i = 0 → S.card = r) → (Set.range a).ncard ≤ 2

number theoryN/A5 sources

declared proved (Lean) · formalized · $100erdos:582

True ↔ ∃ V x G, G.CliqueFree 4 ∧ Erdos582.EdgeRamseyTriangle G

graph theoryramsey theoryN/A5 sources

declared open · formalized · $500erdos:593

Erdos Problem #593 asks which finite triple systems occur in every uncountably chromatic triple system. The answer is exactly the class generated from private-vertex expansions of finite bipartite graphs by finite disjoint unions and one-point amalgamations. Problem #1177 on exact spectra is settled alongside it.

set theorygraph theoryhypergraphsN/A4 sources

declared open · formalizederdos:598

∀ (m : Type u_1) [Infinite m], True ↔ ∃ c, ∀ (X : Set m), Cardinal.mk ↑X = Erdos598.κ → c '' {s | ↑s ⊆ X} = Set.univ

set theoryramsey theoryN/A3 sources

declared disproved (Lean) · formalizederdos:613

False ↔ ∀ n ≥ 3, ∀ (V : Type u_1) [inst : Fintype V] (G : SimpleGraph V) [inst_1 : DecidableRel G.Adj], G.edgeFinset.card = (2 * n + 1).choose 2 - n.choose 2 - 1 → ∃ B D, ∀ [DecidableRel B.Adj] [inst_3 : DecidableRel D.Adj], G = B ⊔ D ∧ B.IsBipartite ∧ ∀ (v : V), D.degree v < n

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:618

True ↔ ∀ (G : (n : ℕ) → SimpleGraph (Fin n)), (∀ (n : ℕ), (G n).CliqueFree 3) → ((fun n => ↑(G n).maxDegree) =o[Filter.atTop] fun n => ↑n ^ (1 / 2)) → (fun n => ↑(Erdos618.h2 (G n))) =o[Filter.atTop] fun n => ↑n ^ 2

graph theoryN/A5 sources

declared solved (Lean) · formalizederdos:619

False ↔ ∃ c > 0, ∀ (V : Type) [inst : Fintype V] (G : SimpleGraph V), G.Connected → G.CliqueFree 3 → ↑(Erdos619.minNewEdges 4 G) < (1 - c) * ↑(Fintype.card V)

graph theoryN/A5 sources

declared proved (Lean) · formalizederdos:621

True ↔ ∀ (n : ℕ) (G : SimpleGraph (Fin n)) (a t : ℕ), IsGreatest {k | ∃ A ⊆ G.edgeFinset, A.card = k ∧ ∀ (x y z : Fin n), G.Adj x y → G.Adj y z → G.Adj x z → ({s(x, y), s(y, z), s(x, z)} ∩ A).card ≤ 1} a → IsLeast {k | ∃ T ⊆ G.edgeFinset, T.card = k ∧ ∀ (x y z : Fin n), G.Adj x y → G.Adj y z → G.Adj x z → (T ∩ {s(x, y), s(y, z), s(x, z)}).Nonempty} t → 4 * (a + t) ≤ n ^ 2

graph theoryN/A5 sources

declared open · formalizederdos:623

Let

X

be a set of cardinality

\aleph_\omega

and

f

a function from the finite subsets of

X

X

such that

f(A)\not\in A

for all

A

. Must there exist an infinite independent

Y\subseteq X

, i.e. with

f(B)\not\in Y

for all finite

B\subset Y

? Claimed resolution: the positive assertion is equivalent to Koepke's free-subset property, hence independent of ZFC, with consistency strength exactly a measurable cardinal.

set theoryN/A4 sources

declared proved (Lean) · formalizederdos:639

True ↔ ∀ᶠ (n : ℕ) in Filter.atTop, ∀ (C : Sym2 (Fin n) → Fin 2), {e | ¬e.IsDiag ∧ ∀ (x y : Fin n), e = s(x, y) → ¬∃ z, z ≠ x ∧ z ≠ y ∧ C s(x, z) = C e ∧ C s(y, z) = C e}.ncard ≤ n ^ 2 / 4

graph theoryramsey theoryN/A5 sources

declared proved (Lean) · formalizederdos:645

∀ (c : ℕ → Bool), ∃ x d, 0 < x ∧ x < d ∧ ∃ C, c x = C ∧ c (x + d) = C ∧ c (x + 2 * d) = C

number theoryadditive combinatoricsramsey theoryN/A5 sources

declared proved (Lean) · formalizederdos:646

True ↔ ∀ (S : Finset ℕ), (∀ p ∈ S, Nat.Prime p) → {n | ∀ p ∈ S, Even (padicValNat p n.factorial)}.Infinite

number theoryfactorialsN/A5 sources

declared verifiable · formalized · £25erdos:647

sorry ↔ ∃ n > 24, ⨆ m, ↑m + (ArithmeticFunction.sigma 0) ↑m ≤ n + 2

number theoryA062249 · A0872804 sources

declared solved (Lean) · formalizederdos:648

(fun n => ↑(Erdos648.g n)) =Θ[Filter.atTop] fun n => √(↑n / Real.log ↑n)

number theoryA3917505 sources

declared disproved (Lean) · formalizederdos:649

False ↔ ∀ (p q : ℕ), Nat.Prime p → Nat.Prime q → ∃ n, n.maxPrimeFac = p ∧ (n + 1).maxPrimeFac = q

number theorypossible5 sources

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