Reiher–LPR extremal function conjecture for triangle-free graphs

For integers $n$ and $s$ with $n/3<s\le n/2$, let $\mathrm{ex}(n,s)$ be the largest number of edges in a triangle-free graph on $n$ vertices whose independence number is at most $s$. For every $k\ge1$, define

$$g_k(n,s)=\frac{k(k-1)n^2}{2}-k(3k-4)ns+\frac{(3k-4)(3k-1)s^2}{2}.$$

Reiher–LPR conjecture. One has

$$\mathrm{ex}(n,s)=\min_k g_k(n,s).$$

This refines Andrásfai's piecewise-quadratic prediction. The paper proves the formula in a neighbourhood to the right of each critical ratio, while the asserted formula over the full range remains open.