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Claim. There is an absolute constant $c$ such that every graph with average degree at least $c$, and so every graph with minimum degree at least $c$, contains a cycle of length $2^k$ for some $k\ge2$. The result is Corollary 1.3 of H. Liu and R. Montgomery, A solution to Erdős and Hajnal's odd cycle problem, J. Amer. Math. Soc. 36 (2023), 1191--1234, first posted as arXiv:2010.15802 on 2020-10-29 (the claim's date): there is $d_0$ such that for every increasing sequence $(\sigma_i)_{i\ge1}$ of positive even integers with $\sigma_{i+1}\le\exp(\sigma_i^{1/10})$, every graph with average degree at least $\max\{d_0,\sigma_1^2\}$ contains a cycle of length $\sigma_i$ for some $i$. The corpus states the corollary on its result page, deduced there from the even-cycle interval theorem, Theorem 1.1. Since $\log(2x)=o(x^{1/10})$, the powers of two satisfy the growth condition from some index $i_0\ge2$ on, and the tail $(2^i)_{i\ge i_0}$ gives the claim with $c=\max\{d_0,4^{i_0}\}$. The same corollary is the second accepted claim of Problem 72.

Covers. The statement of Problem 64 for every graph whose average degree is at least $\max\{d_0,4^{i_0}\}$, in particular for every graph of minimum degree at least that constant; the cycle found has length $2^k$ with $k\ge i_0\ge2$. Neither $d_0$ nor $i_0$ is made explicit in the paper, and the threshold is far above $3$, so graphs of minimum degree between $3$ and the constant are not covered. The site's remark credits the paper with the affirmative answer once the minimum degree exceeds an absolute constant, which refutes the stronger expectation of Erdős and Gyárfás that for every $r$ some graph of minimum degree $r$ avoids all such cycles.

Depends on. No page of this wiki: the corollary is the paper's own, recorded on its library result page.

Acceptance. Refereed: the paper is a publication in the Journal of the American Mathematical Society (published online 2023-03-31). The site's curator credits the paper with the case of large minimum degree while labeling the problem FALSIFIABLE, so the curator's remark is commentary on an open problem and is not listed as reviewed evidence. The corpus's source card reconstructs the proof from the arXiv v2 manuscript; that reconstruction is incomplete at Lemma 3.13's final reservoir compatibility, where the selected path is not shown to avoid earlier selected reservoirs, and it is author-recorded, not independently reviewed. This concerns the compilation and not the published result; the acceptance recorded here rests on the publication.