Given a graph G, a Berge copy of G is a hypergraph obtained by enlarging the edges arbitrarily. Gyori in 2006 showed that for r=3 or r=4, an r-uniform n-vertex Berge triangle-free hypergraph has at most floor[n^2/8(r-2)] 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, Győri, Nagy-György, Paulos, Xiao and Zamora 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 floor[n^2/8].
We prove this conjecture for t=2 and disprove it for t>2 by proving the sharp bound floor[n^2/8]+(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.
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