For n≥30(s+1), every F5-free 3-graph with matching number at most s has at most s⌊(n−s)^2/4⌋ edges, with equality only for the complete 3-partite hypergraph H3(n,s).
Extremal results for graphs avoiding a rainbow subgraph
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We say that $k$ graphs $G_1,G_2,\dots,G_k$ on a common vertex set of size $n$ contain a rainbow copy of a graph $H$ if their union contains a copy of $H$ with each edge belonging to a distinct $G_i$. We provide a counterexample to a conjecture of Frankl on the maximum product of the sizes of the edge sets of three graphs avoiding a rainbow triangle. We propose an alternative conjecture, which we prove under the additional assumption that the union of the three graphs is complete. Furthermore, we determine the maximum product of the sizes of the edge sets of three graphs or four graphs avoiding a rainbow path of length three.
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Hypergraph Tur\'an problem of the generalized triangle with bounded matching number
For n≥30(s+1), every F5-free 3-graph with matching number at most s has at most s⌊(n−s)^2/4⌋ edges, with equality only for the complete 3-partite hypergraph H3(n,s).