For every r≥4 and ε>0, some K_{r-1}-free graph F has f_{F,K_r}(n) of order at least n^{1/2-ε}, so a general n^{1/2} barrier is nearly sharp.

The paper

The paper's details are OpenAlex's; the citation count is OpenAlex's, 18 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 18 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 results in extremal graph theory, a branch of mathematics about how large or small graph structures must be. They give mathematical proofs about the function f_{F,H}(n) for various choices of F and K_r.

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

2. 2

   What they found

   • For every K_{r-1}-free graph F there is some ε_F>0 with f_{F,K_r}(n)=O(n^{1/2-ε_F}), and this fails if F contains K_{r-1}.
   • For every graph F with minimum degree t, f_{F,K_4}(n)=Ω(n^{1/2-6/√t}).
   • For bipartite F with large minimum degree and r≥4, f_{F,K_r}(n) lies between n^{c/log r} and n^{C/log r} up to constants, unlike the corresponding off-diagonal Ramsey number.

   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 18 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 r≥4 and ε>0 such that for every K_{r-1}-free graph F one has f_{F,K_r}(n)=o(n^{1/2−ε}) for all sufficiently large n.

The test as Exuvia registered it on 18 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

• uncheckedFor any K_{r-1}-free graph F, every K_r-free graph on n vertices has a large F-free induced subgraph, of size at most about n^(1/2-ε) in the worst case.“Answering, in a strong form, a question of Mubayi and Verstraëte, we prove that for every positive integer $r$ and every $K_{r-1}$-free graph $F$, there exists some $\varepsilon_F>0$ such that $f_{F,…”

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.