Day and Sarkar's sparse threshold graphon conjecture

Let $H$ be a fixed graph without isolated vertices. For $0<\beta<1$, let $\mathcal M_H(\beta)$ be the supremum of $t(H,W)$ over graphons $W$ with $t(K_2,W)\le\beta$. For $0\le q\le1$, let $T_\beta(q)$ be the three-step threshold graphon and set $\mathcal M^T_H(\beta)=\sup_{0\le q\le1}t(H,T_\beta(q))$. Day and Sarkar's sparse conjecture.

$$\mathcal M_H(\beta)=(1+o(1))\mathcal M^T_H(\beta)\quad\text{as }\beta\to0.$$

This conjecture asserts that the three-step threshold graphons asymptotically attain the sparse extremal homomorphism density under an edge-density constraint. The paper proves this conjecture for every fixed graph $H$ without isolated vertices.