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Claim. The wording of Problem 1077 is false, and the size of the almost-regular subgraph it should have asked for is $n^\alpha$. The witness: the complete bipartite graph $K_{a,n-a}$ with $a\approx n^\alpha$ has about $n^{1+\alpha}$ edges, and every $D$-balanced subgraph of it with an edge has at most $(D+1)a$ vertices, because each of its edges has an end on the small side and the degree bounds cap the large side at $D$ times the small side. For $0<\alpha<\tfrac12$ that bound is below $n^{1-\alpha}$ for large $n$, so no $D$-balanced subgraph on more than $n^{1-\alpha}$ vertices has an edge, for every $\epsilon>0$ and every $D$; since the question quantifies over every $\epsilon,\alpha>0$, this refutes it as a whole, and the problem page recomputes the count with $a=\lceil2n^\alpha\rceil$. The same graph shows that in any question of this shape the subgraph can have at most a constant times $n^\alpha$ vertices. The comment itself states the exponent and the two bounds and does not say that it refutes the printed wording; reading it as the disproof of the exponent $1-\alpha$ is the site's acceptance (the curator's reply of 31 December 2025 and the commentary) and the problem page's own count. The comment's second part, not part of the disproof, is a lower bound of the same order: Theorem 1.3 of the preprint of Jiang and Longbrake (arXiv:2507.03261) gives a $6$-almost-regular subgraph of average degree of order $n^\alpha/\log n$, and the comment reports that its proof, followed carefully, gives at least $n^\alpha/2$ vertices. The problem page records this lower bound as resting on a preprint and a forum reading of its proof, with the logarithm unresolved on the sources read there.

Depends on. Nothing in this wiki; the count is elementary.

Acceptance. Reviewed: the site's curator, Thomas Bloom, adopted the comment (marked on the thread as addressed by a site update, the page last edited 8 January 2026): the commentary credits JunGao with the exponent $\alpha$, names the complete bipartite graph as the witness for the upper bound, and states its best guess at the intended question, with $n^\alpha$; the site labels the problem DISPROVED (LEAN), Bloom took no part in the comment, and the community database lists the problem's informal status as disproved as of its last update on 29 December 2025. Not refereed: the comment is a forum post, and the preprint it relies on for the lower bound had no journal record on 2026-09-18. The accepted claim is the refutation of the Statement; the answer to the curator's variant, recorded in the problem page's Formulation, is not a claim here. The earlier clique counterexample has its own claimed page, clique counterexample, and a Lean file refuting the formal statement with this witness family at $\alpha=\tfrac14$ has the page Lean file. The acceptance recorded here rests on the site's acceptance and the elementary count, not on an independent review.