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Random Turán Problems for $K_{s,t}$ Expansions (2412.09367v1)
Published 12 Dec 2024 in math.CO and math.PR
Abstract: Let $K_{s,t}{(r)}$ denote the $r$-uniform hypergraph obtained from the graph $K_{s,t}$ by inserting $r-2$ new vertices inside each edge of $K_{s,t}$. We prove essentially tight bounds on the size of a largest $K_{s,t}{(r)}$-subgraph of the random $r$-uniform hypergraph $G_{n,p}r$ whenever $r\ge 2s/3+2$, giving the first random Tur\'an results for expansions that go beyond a natural "tight-tree barrier." In addition to this, our methods yield optimal supersaturation results for $K_{s,t}{(3)}$ for sufficiently dense host hypergraphs, which may be of independent interest.