Erdős (Erdos) Problem #161: status in 2026

Erdős Problem #161

Let α∈[0,1/2) and n,t≥ 1. Let F^(t)(n,α) be the smallest m such that we can 2-colour the edges of the complete t-uniform hypergraph on n vertices such that if X⊆ [n] with lvert Xrvert ≥ m then there are at least α C(lvert Xrvert, t) many t-subsets of X of each colour.

Status (as of 2026-11-11): Open according to erdosproblems.com (status last updated 2025-08-31). Erdős offered a prize of $500.

Current state (tracked automatically)

erdosproblems.com lists Erdős problem #161 as open (status updated 31 August 2025).

All tracked facts and sources

Fact	Value	Source
Status of Erdős problem #161 on erdosproblems.com	open 2025-08-31	erdosproblems.com
Formal Conjectures: research statements in the file	2 open, 0 solved	Formal Conjectures

Read twice a day from public sources by deterministic collectors (no AI; the sentences above are filled-in templates): erdosproblems.com (checked 2026-11-12), Formal Conjectures (checked 2026-11-12). How it works.

From the catalogue. Imported from The Formal Conjectures Authors (Google DeepMind and contributors) (Apache-2.0) — original. Nobody has started on it here yet: tasks are created as soon as someone asks for one or submits a claim.

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Cite

@misc{cairn-erdos-161,
  title        = {Erdős Problem #161},
  author       = {{Cairn Commons contributors}},
  howpublished = {\url{https://cairn-commons.com/problems/erdos-161}},
  year         = {2026},
  note         = {Open problem on Cairn Commons, CC BY 4.0. Accessed 2026-11-12}
}

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Claims

Verified

Disputed

Refuted

On the literature board

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What a result looks like: C-16, a newcomer working with an AI model found C(85,24,3) ≤ 76 (previous best 77), and a computer verified it within minutes. A documented dead end counts too.

The problem

The question

erdos_161.parts.i. Let $α \in [0, 1/2)$ and $n, t \geq 1$ . Let $F^{(t)} (n, α)$ be the smallest $m$ such that we can $2$ -colour the edges of the complete $t$ -uniform hypergraph on $n$ vertices such that if $X\subseteq [n] $w i t h$ \lvert X\rvert \geq m $t h e n t h er e a r e a tl e a s t$ \alpha \binom{\lvert X\rvert}{t}$ many $t$ -subsets of $X$ of each colour. For fixed $n, t$ as we change $α$ from $0$ to $1/2$ does $F^{(t)} (n, α)$ increase continuously or are there jumps? Only one jump?

Here growth is compared up to constant factors as $n \to \infty$ . At $α = 0$ , each color must occur, following the source's Ramsey convention.

erdos_161.parts.ii. For fixed $t \geq 3$ , do all positive densities below $1/2$ give the same asymptotic order?

Formal statement (Lean 4)

From Formal Conjectures, module FormalConjectures.ErdosProblems.«161» (2 statements).

theorem erdos_161.parts.i :
    ∀ t : ℕ, 1 ≤ t →
      {p | p.1 ∈ Set.Ico (0 : ℝ) (1 / 2) ∧ p.2 ∈ Set.Ico (0 : ℝ) (1 / 2) ∧
        (fun n ↦ (Hypergraph.balancedColoringThreshold n t p.1 : ℝ)) =Θ[atTop]
          (fun n ↦ (Hypergraph.balancedColoringThreshold n t p.2 : ℝ))} =
        erdos_161.parts.i.answer t

theorem erdos_161.parts.ii :
    ∀ t : ℕ, 3 ≤ t → ∀ α β : ℝ,
    0 < α → α < 1 / 2 → 0 < β → β < 1 / 2 →
    (fun n ↦ (Hypergraph.balancedColoringThreshold n t α : ℝ)) =Θ[atTop]
      (fun n ↦ (Hypergraph.balancedColoringThreshold n t β : ℝ))

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/161. Report it as a literature claim.
A precise flaw in the formal statement (a misformalisation) — report it upstream too.

References

erdosproblems.com/161
[CFS11] Conlon, David and Fox, Jacob and Sudakov, Benny, Large almost monochromatic subsets in hypergraphs. Israel J. Math. (2011), 423--432.
[Er90b] Erdős, Paul, Problems and results on graphs and hypergraphs: similarities and differences . Mathematics of Ramsey theory (1990), 12-28.

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.