The Turán Number of Berge Book Hypergraphs

Given a graph \(G\), a Berge copy of \(G\) is a hypergraph obtained by enlarging the edges arbitrarily. Győri [Combin. Probab. Comput., 15 (2006), pp. 185–191] showed that for \(r=3\) or \(r=4\), an \(r\)-uniform \(n\)-vertex Berge triangle-free hypergraph has at most \(\lfloor n^2/8(r-2)\rfloor\) hyperedges if \(n\) is large enough, and this bound is sharp. The book graph \(B_t\) consists of \(t\) triangles sharing an edge. Very recently, Ghosh et al. [Discrete Math., 347 (2024), 113828] showed that a 3-uniform \(n\)-vertex Berge \(B_t\)-free hypergraph has at most \(n^2/8+o(n^2)\) hyperedges if \(n\) is large enough. They conjectured that this bound can be improved to \(\lfloor n^2/8\rfloor\). We prove this conjecture for \(t=2\) and disprove it for \(t\gt 2\) by proving the sharp bound \(\lfloor n^2/8\rfloor +(t-1)^2\). We also consider larger uniformity and determine the largest number of Berge \(B_t\)-free \(r\)-uniform hypergraphs besides an additive term \(o(n^2)\). We obtain a similar bound if the Berge \(t\)-fan (\(t\) triangles sharing a vertex) is forbidden.