A note on the uniformity threshold for Berge hypergraphs (Q2145761)

A Berge copy of a graph is a hypergraph obtained by enlarging the edges arbitrarily. \textit{D. Grósz} et al. [Eur. J. Comb. 88, Article ID 103109, 9 p. (2020; Zbl 1442.05146)] showed that for any graph \(F\), there is an integer \(r_0 = r_0(F )\), such that for any \(r \geq r_0\), any \(r\)-uniform hypergraph without a Berge copy of \(F\) has \(o(n^2)\) hyperedges. The smallest such \(r_0\) is called the uniformity threshold of \(F\) and is denoted by th(\(F\) ). They showed that th(\(F ) \leq R(F , F^\prime)\), where \(R\) denotes the off-diagonal Ramsey number and \(F'\) is any graph obtained from \(F\) by deleting an edge. The author improves this bound to th(\(F ) \leq R(K_{\chi(F )}, F^\prime)\), and uses the new bound to determine th(\(F\) ) exactly for several classes of graphs: the \(k\)-fan, consisting of \(k\) triangles sharing a vertex, the book with \(t\) pages, which consists of \(t\) triangles sharing an edge, the wheel with \(k\) spokes, and the generalized book, formed by \(m\) copies of \(K_q\) each sharing a fixed set of \(p\) vertices. Note that in this paper the author also initiated the study of generalized cover Turán numbers for Berge hypergraphs.