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Claim. The wording of Problem 1077 is false. Fix $0<\alpha<\tfrac13$ and let $G$ be a clique on $k=\lceil\sqrt2\,n^{(1+\alpha)/2}\rceil+1$ vertices together with $n$ vertices in all, the rest isolated; then $G$ has $k(k-1)/2>n^{1+\alpha}$ edges. The comment itself takes a clique on slightly more than $n^{(1+\alpha)/2}$ vertices and says that it has more than $n^{1+\alpha}$ edges; that size gives only about $n^{1+\alpha}/2$ edges, and the factor $\sqrt2$ here repairs the count without changing the argument. A $D$-balanced subgraph with an edge lies inside the clique, since a vertex of degree zero beside an edge breaks the balance condition, and the clique has fewer than $n^{1-\alpha}$ vertices for large $n$ because $(1+\alpha)/2<1-\alpha$; so no $D$-balanced subgraph on more than $n^{1-\alpha}$ vertices has an edge, for every $\epsilon>0$, every $D$ and every large $n$. A second comment of 30 December 2025 says, without argument, that putting the other vertices on a path still refutes the question. The reason, this page's own deduction and not the comment's: a $D$-balanced subgraph with a vertex on the path has minimum degree at most $2$, so at most $D$ times its vertex count in edges, below $\epsilon m^{1+\alpha}$ for large $m$, while one inside the clique has too few vertices as before. Both comments are Boris Alexeev's, and Alexeev credits both examples to Aristotle. Since the question quantifies over every $\epsilon,\alpha>0$, one $\alpha$ refutes it as a whole; the examples decide nothing about the question restricted to $\alpha\ge\tfrac13$, and the curator's guess at the intended question, with $n^\alpha$ in place of $n^{1-\alpha}$, is a variant recorded in the problem page's Formulation.

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

Standing. Claimed. The site's label DISPROVED (LEAN) and its commentary (last edited 8 January 2026) credit JunGao and the complete bipartite witness of the claim page JunGao, and its thanks line names Yael Dillies, JunGao and Zach Hunter; neither names this comment, Alexeev or Aristotle, so the site's acceptance does not cover this claim and the page lists no reviewed evidence. The curator's thread replies are discussion, not acceptance: on 25 December 2025 the curator guessed that the authors meant graphs without isolated vertices and asked for an expert's view, and on 31 December 2025, after the path variant, conceded the point and wrote that the exponent $1-\alpha$ was a typo and that the intended question was the one JunGao answered. Not refereed: the examples exist only as forum comments. The problem page recomputes the complete bipartite witness and not this one; the count above is the comment's argument with its clique size corrected.