The Andr\e1sfai blow-up conjecture for triangle-free graphs

Let $n$ and $s$ be integers with $n s\e 0$. For $k\e 2$, define

$$g_k(n,s)=\tfrac12 k(k-1)n^2-k(3k-4)ns+\tfrac12(3k-4)(3k-1)s^2.$$

Define $g(n,s)$ by

$$g(n,s)=\begin{cases} \frac12ns,&s\le \frac13n,\\ g_k(n,s),&\frac{k}{3k-1}n\le s<\frac{k-1}{3k-4}n\text{ for some }k\ge2,\\ \left\lfloor\frac{n^2}{4}\right\rfloor,&\frac n2\le s\le n. \end{cases}$$

Here $\operatorname{ex}(n,s)$ denotes the maximum number of edges in a triangle-free $n$-vertex graph whose independence number is at most $s$. Andr\e1sfai blow-up conjecture. For all integers $n\ge s\ge0$, $\operatorname{ex}(n,s)\le g(n,s)$. Equivalently, every triangle-free $n$-vertex graph $G$ with $\alpha(G)\le s$ has at most $g(n,s)$ edges. The conjecture is known for the ranges governed by $g_2$ and $g_3$, and is open only for $s\in(\frac13n,\frac38n)$. Its proposed bound is attained, for the most interesting range of $s/n$, by suitable blow-ups of Andr\e1sfai graphs.