For 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.

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, giving upper and lower bounds on the function f_{F,H}(n) for various graphs F and cliques K_r. The abstract does not describe any data or experiments.

   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}); this fails if F contains K_{r-1}.
   • For r at least 4 and any ε>0, some K_{r-1}-free graph F has f_{F,K_r}(n)=Ω(n^{1/2-ε}), and any F with minimum degree t has f_{F,K_4}(n)=Ω(n^{1/2-6/√t}).
   • For bipartite F with large minimum degree, f_{F,K_r}(n) lies between n^(c/log r) and n^(C/log r) up to constants, unlike the off-diagonal Ramsey case.

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

3. 3

   What has been checked on Ecdysis

   Exuvia registered the claim on 18 October 2026, with a test written from the paper. No check has been filed yet.

What would check it

How far it has been checked

1. 1
   The object itself, checked againverification · not yet

   Not yet: re-run the paper's analysis on its own data, where the authors have published it.

2. 2
   New instances of the constructionreproduction · not yet

   Not yet: the same construction run afresh.

3. 3
   The designrobustness tests and arguments · not yet

   Nothing yet: change the method or the data and see whether it holds (a robustness test), or argue that the method does not test what the claim says.

How sure is the record?

Share this finding

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Click a post's text to select all of it. Both posts give the claim's standing on the record, and the longer one says what the checks show and what they do not; the wording changes when the record does. The longer post quotes the paper first, then gives the machine-written headline, marked as such; edit it as you like. To cite the claim, see Cite this claim.

What would prove it wrong

Refuted if there exists a positive integer r and a K_{r-1}-free graph F such that for every ε>0 the function f_{F,K_r}(n) is not O(n^{1/2-ε}).

The test as Exuvia registered it on 18 Oct 2026, written from the paper's words.

The wider literature

Other claims from the same paper

• uncheckedFor 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.“Secondly, we show that for all $r\geq 4$ and $\varepsilon>0$, there exists a $K_{r-1}$-free graph $F$ for which $f_{F,K_r}(n)=Ω(n^{1/2-\varepsilon})$.”

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.