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Claim. Results of A. A. Razborov, More about sparse halves in triangle-free graphs, Mat. Sb. 213 (2022), no. 1, 119--140 (English translation Sb. Math. 213 (2022), no. 1), first posted as arXiv:2104.09406 on 2021-04-19 and cited as [Ra22] on the problem page. The paper writes $\beta(G)$ for the least number of edges spanned by half the vertices of $G$ divided by $n^2$, $\rho(G)=2e(G)/n^2$ for the edge density and $\alpha(G)$ for the independence number divided by $n$, and calls Erdős's conjecture $\beta(G)\le1/50$ for every triangle-free $G$ Conjecture 1. Theorem 3.2: $\beta(G)\le27/1024$ for every triangle-free $G$. Conjecture 1 holds for every triangle-free graph without an induced matching of size $2$ (Theorem 3.3); with $\rho(G)\le(33-\sqrt{161})/116\approx0.1751$, that is, with at most $(33-\sqrt{161})n^2/232$ edges (Theorem 3.4); that is strongly regular (Theorem 3.5, first stated in the arXiv version of 2021-07-28); with $\alpha(G)\ge2/5$ (Corollary 3.7, from Theorem 3.6's bound $\beta(G)\le\frac12\alpha(G)(\frac12-\alpha(G))$ for $\alpha(G)\ge3/8$), hence also with maximum degree at least $2n/5$, a vertex's neighborhood being independent; and of girth at least $5$ (Theorem 3.8). Each is the contrapositive of Problem 128 on its class, and the low-density case extends Keevash and Sudakov's range of at most $n^2/12$ edges. The $27/1024$ bound is the limit of the method that weights the three parts cut out by one edge uniformly, with the Clebsch graph extremal, and it alone settles no instance of the problem. Library home razborov_2022_more_about_sparse_halves_triangle_free.

Covers. Triangle-free graphs without an induced matching of size $2$; of girth at least $5$; with independence number at least $2n/5$ (so with maximum degree at least $2n/5$); with edge density $\rho(G)\le(33-\sqrt{161})/116$; and strongly regular. Not covered: the general question, for which the paper gives the constant $27/1024$ in place of $1/50$, which settles no instance; the question as posed stays open.

Depends on. Nothing in this wiki; Proposition 1.1 reproves the part of Krivelevich's argument it uses, and the flag-algebra inequality of Theorem 3.1 is proved in the paper with a Maple worksheet the author posts.

Acceptance. Refereed: the paper appeared in Matematicheskii Sbornik with its English translation in Sbornik: Mathematics. The site's commentary credits the paper with the constant $27/1024$ only and labels the problem FALSIFIABLE, which settles nothing, so that credit is not listed as reviewed.