For t−2 ≥ s ≥ 4, the Erdős–Rogers function f_{s,t}(n) is bounded above by a constant times the square root of n.

The paper

The paper's details are OpenAlex's; the citation count is OpenAlex's, 23 Oct 2026. The line on the paper is machine-written, as noted under Why it matters.

Why it matters

Written by Claude (claude-sonnet-5-5) on 23 Oct 2026 from the paper's abstract (as arXiv publishes it) and its OpenAlex record. Machine-written context to help a reader: it is not evidence, it moves no number, and it may be wrong. The quoted sentence is the claim; where it stands is computed from the record. If it misreads the paper, tell the stewards.

The story so far

1. 1

   What the authors did

   The authors prove upper bounds on the Erdős–Rogers function by mathematical argument, and use them to partially answer a question of Erdős. The abstract does not describe the methods further.

   Machine-written from the paper's abstract, as noted under Why it matters.

2. 2

   What they found

   • For every s ≥ 3, f_{s,s+1}(n) ≤ c_1 (log n)^{c_2} √n, which the authors say is best possible up to a polylogarithmic factor.
   • For all t−2 ≥ s ≥ 4, there is a constant c_3 with f_{s,t}(n) ≤ c_3 √n.
   • The ratio f_{s+1,s+2}(n) / f_{s,s+2}(n) tends to infinity for any s ≥ 4, partially answering a question of Erdős.

   Machine-written from the paper's abstract, as noted under Why it matters.

3. 3

   What has been checked on Ecdysis

   Exuvia registered it on 23 October 2026. Its credence, the record's estimate that it holds, is 0.55 on a scale from 0 (refuted) to 1 (established): where it started, as every claim from the literature does. Only independent evidence moves it.

What would check it

How sure is the record?

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What would prove it wrong

Refuted if there exist integers s, t with t-2 ≥ s ≥ 4 such that f_{s,t}(n) / √n is unbounded as n → ∞ (i.e., for every constant C there are arbitrarily large n with f_{s,t}(n) > C√n).

The test as Exuvia registered it on 23 Oct 2026, written from the paper's words. A conceptual claim's test names its refuter in words: it is checked by argument.

The wider literature

Other claims from the same paper

• uncheckedIn this paper, we show that for every $s\ge 3$ there exist constants $c_1=c_1(s)$ and $c_2=c_2(s)$ such that $f_{s,s+1}(n) \le c_1 (\log n)^{c_2} \sqrt{n}$.
• uncheckedFor every s at least 4, the ratio f_{s+1,s+2}(n) to f_{s,s+2}(n) grows without bound as n grows, partially answering a question of Erdős.“In doing so, we partially answer a question of Erdős by showing that $\lim_{n\to \infty} \frac{f_{s+1,s+2}(n)}{f_{s,s+2}(n)}=\infty$ for any $s\ge 4$.”

Headlines are machine-written from the paper's abstract, or from the quote and the paper's title where no abstract is open; each claim's own words are quoted beneath its headline.