The Tur\'an number of Berge book hypergraphs

TL;DR

This paper determines the maximum number of hyperedges in large Berge $B_t$-free hypergraphs, confirming a conjecture for $t=2$, disproving it for $t>2$, and extending results to larger uniformities and related structures.

Contribution

It proves the sharp bound for Berge $B_t$-free hypergraphs, confirming the conjecture for $t=2$ and providing counterexamples for $t>2$, also extending to larger uniformities and other forbidden configurations.

Findings

Confirmed the conjecture for $t=2$ with the bound $loor{n^2/8}+(t-1)^2$.

Disproved the conjecture for $t>2$ by establishing the bound $loor{n^2/8}+(t-1)^2$.

Extended the bounds to larger uniformities and related forbidden structures.

Abstract

Given a graph $G$, a Berge copy of $G$ is a hypergraph obtained by enlarging the edges arbitrarily. Gy\H ori in 2006 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, Gy\H{o}ri, Nagy-Gy\"orgy, 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 $\lfloor n^2/8\rfloor$. We prove this conjecture for $t=2$ and disprove it for $t>2$ 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.