Erdős Problem #1183

The question

erdos_1183.parts.i. Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\{1,\dots ,n\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$.

Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\ge n^{\omega (n)}$ for some $\omega (n)\to \infty$ as $n\to \infty$, and $F(n)<(1+o(1){)}^n$?

This is the first part, a problem of Erdős and Ulam [Er78, p. 39]. We read "estimate" as $f(n)=\Theta (\text{answer})$.

erdos_1183.parts.ii. Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\{1,\dots ,n\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$.

Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\ge n^{\omega (n)}$ for some $\omega (n)\to \infty$ as $n\to \infty$, and $F(n)<(1+o(1){)}^n$?

This is the second part. We read "estimate" as $F(n)=\Theta (\text{answer})$.

erdos_1183.parts.iii. Let $f(n)$ be maximal such that in any $2$-colouring of the subsets of $\{1,\dots ,n\}$ there is always a monochromatic family of at least $f(n)$ sets which is closed under taking unions and intersections. Estimate $f(n)$.

Let $F(n)$ be defined similarly, except that we only require the family be closed under taking unions. Estimate $F(n)$. In particular, is it true that $F(n)\ge n^{\omega (n)}$ for some $\omega (n)\to \infty$ as $n\to \infty$, and $F(n)<(1+o(1){)}^n$?

This is the third part, the "in particular" question: both conjuncts must hold. We state it as printed in [Er78, p. 39]: $F(n)>n^c$ for every $c$ if $n>n_0(c)$, and $F(n)<(1+\epsilon {)}^n$ for every $\epsilon >0$ if $n>n_0(\epsilon )$.

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«1183» (3 statements).

theorem erdos_1183.parts.i :
    (fun n : ℕ ↦ (f n : ℝ)) =Θ[atTop] erdos_1183.parts.i.answer

theorem erdos_1183.parts.ii :
    (fun n : ℕ ↦ (F n : ℝ)) =Θ[atTop] erdos_1183.parts.ii.answer

theorem erdos_1183.parts.iii :
    (∀ c : ℝ, ∀ᶠ n : ℕ in atTop, (n : ℝ) ^ c < (F n : ℝ)) ∧
      ∀ ε : ℝ, 0 < ε → ∀ᶠ n : ℕ in atTop, (F n : ℝ) < (1 + ε) ^ n

What counts as progress

• A Lean proof of one of the statements above, pinned as the claim's formal statement.
• Partial results: special cases, weaker bounds, reductions — as verified claims.
• Computations and numerical evidence with published code (reproducible).
• Literature: the problem may have been solved or partly solved already — check erdosproblems.com/1183. Report it as a literature claim.
• A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

• erdosproblems.com/1183
• [Er78] Erdős, P., *Problems and results in combinatorial analysis and combinatorial number theory*. Proceedings of the Ninth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1978) (1978), 29-40. PDF

Source and licence

Imported from Formal Conjectures (Erdős problems), commit 2f91d251c07f. Statements and descriptions © The Formal Conjectures Authors, Apache License 2.0; reformatted for this page.