Erdős–Hajnal conjecture for the iterated-blowup threshold

Let $g_k(n)$ be the maximum number of edges in an $n$-vertex iterated blowup of an edge. Thus $g_k(n)=0$ for $n<k$, and for $n\geq k$,

$$g_k(n)=\max_{s_1+\cdots+s_k=n}\left(\prod_{j=1}^k s_j+\sum_{j=1}^k g_k(s_j)\right),$$

where the maximum is over compositions of $n$ into $k$ positive integer parts. Erdős and Hajnal showed that $r_k(s,g_k(s);t)=t^{O(1)}$. Erdős–Hajnal conjecture. For all $s>k\geq 3$,

$$r_k(s,g_k(s)+1;t)=2^{t^{\Omega(1)}}.$$

This asserts that one additional red edge beyond the iterated-blowup threshold changes the Ramsey number from polynomial growth to exponential growth in a positive power of $t$. The paper states that its main theorem settles this conjecture, although the optimal exponent of $t$ in the exponential remains open.