Korándi–Roberts–Scott conjecture on blowups of odd-cycle-free graphs

Fix $k\ge 2$. Let $\delta>\delta_0>0$ and let $n$ be sufficiently large. For a graph $G$, write $D_2(G)$ for the maximum number of edges in a bipartite subgraph of $G$, and call a graph obtained by replacing vertices of a graph with independent sets according to its adjacency pattern a blowup. Korándi–Roberts–Scott conjecture. For every $C_{2k-1}$-free graph $G$ on $n$ vertices satisfying

$$(1/4-\delta_0)n^2\ge e(G)\ge(1/4-\delta)n^2,$$

there is a $C_{2k+1}$-blowup $G^*$ such that

$$e(G^*)\ge e(G)\quad\text{and}\quad D_2(G^*)\ge D_2(G).$$

This conjecture concerns the structure of graphs with edge density close to one quarter and is presented as a way to extend the range of near-extremal results for odd-cycle-free graphs. No resolution is given in the supplied text.