As the site words it, $h(n)$ is the least threshold that every such graph meets, and every graph meets the threshold $1$ (a single vertex is a clique), so $h(n)\le1$ for every $n\ge1$ and the first displayed inequality $n^{1/3+c_1}\ll h(n)$ fails at every $n$; the question would then have the trivial answer no. The check is the corpus's own. The change replaces the single word "minimal" with "maximal", so that $h(n)$ is the largest clique size guaranteed in every such graph, the minimum of the clique number over them. The evidence: Bucić and Sudakov, stating the Erdős--Hajnal question, are "interested in the smallest possible size of $\alpha(G)$ in an $n$-vertex graph satisfying $\alpha_m(G)\geq r$" ([BuSu23], p. 2 of arXiv v3), name that quantity $f(n,m,r)$ (p. 5), and report that Erdős and Hajnal "observed that any graph $G$ on $n$ vertices with $\alpha_7(G)\geq 3$ must have $\alpha(G)\geq\Omega(n^{1/3})$" and that some such graph has $\alpha(G)\le O(n^{1/2})$ (p. 2); in the complement these are bounds on the largest clique forced, which is $h(n)$ only with "maximal". The site's own commentary credits Erdős and Hajnal with $n^{1/3}\ll h(n)\ll n^{1/2}$ and Bucić and Sudakov with $h(n)\gg n^{5/12-o(1)}$, bounds true only of the corrected form, and keeps the label OPEN, which only the corrected form fits; the word "maximal" is the form the site uses for the neighboring question of Erdős and Hajnal, Problem 804. The poser's own text, Erdős's 1991 paper [Er91], is cited here only through [BuSu23] and the site, so whether the slip is the site's or already in [Er91] is not known. No result about the site's wording is recorded.