Erdős–Hajnal–Simonovits–Sós–Szemerédi periodicity conjecture for Ramsey–Turán extremal graphs

Let $p>2$ and let $\varrho_p(q)$ denote the Ramsey–Turán density for $K_q$-free graphs with independence number $o(n)$. Write

$$q=pt+r+2,$$

where $t\in\mathbb{N}$ and $0\le r<p$. An asymptotic extremal graph $G$ is a graph sequence on $n$ vertices attaining $\varrho_p(q)$ asymptotically, and $d_G(A,B)$ denotes the edge-density between vertex sets $A$ and $B$.

Erdős–Hajnal–Simonovits–Sós–Szemerédi conjecture. The asymptotic extremal graphs $G$ for $\varrho_p(q)$ have a partition

$$V(G)=V_0\cup V_1\cup\cdots\cup V_t$$

with $e(G[V_i])=o(n^2)$ for every $0\le i\le t$, with

$$d_G(V_0,V_1)=\frac{r+1}{p}-o(1),$$

the degrees in $G[V_0,V_1]$ differing by $o(n)$, and

$$d_G(V_i,V_j)=1-o(1)$$

for all pairs $\{i,j\}\ne\{0,1\}$. In particular,

$$\varrho_p(q)=\varrho_p^*(q):=\frac{(t-1)(2p-r-1)+r+1}{t(2p-r-1)+r+1}.$$

This conjecture predicts periodic dependence of the Ramsey–Turán density and the corresponding asymptotic extremal structure on the residue of $q$ modulo $p$. The case $p=2$ is known, while the general case was largely open when proposed; the paper specifically identifies the determination of $\varrho_3(5)$ as a previously open problem.