Logarithm-free inducibility bound conjecture

For an $n$-vertex graph $G$, let $X_{G,k}$ be the number of edges induced by a uniformly random $k$-vertex subset of $G$. Define

$$I(n,k,\ell)=\max\{\Pr(X_{G,k}=\ell):|V(G)|=n\}$$

and

$$\operatorname{ind}(k,\ell)=\lim_{n\to\infty}I(n,k,\ell).$$

For $0\le \ell\le\binom{k}{2}$, define

$$\ell^*=\min\mathopen{}\mathclose\bgroup\originalleft\{\ell,\binom{k}{2}-\ell\aftergroup\egroup\originalright\}.$$

Logarithm-free inducibility bound conjecture. For all $k$ and all $0\le\ell\le\binom{k}{2}$, we have

$$\operatorname{ind}(k,\ell)=O\mathopen{}\mathclose\bgroup\originalleft(\sqrt{k/\ell^*}\aftergroup\egroup\originalright).$$

The conjecture is proposed to remove the logarithmic factor appearing in the paper's preceding bound, whose logarithm arises from an estimate related to the Gotsman–Linial conjecture. Its status is not resolved in the supplied text.