Extremal Numbers of Hypergraph Suspensions of Even Cycles

TL;DR

This paper extends classical bipartite cycle-free graph problems to hypergraph suspensions, establishing bounds on the maximum size of hypergraphs avoiding certain cycles and constructing dense cycle-free bipartite graphs.

Contribution

It proves the order of magnitude for the maximum number of triples in hypergraphs avoiding a specific cycle and constructs new dense bipartite graphs free of that cycle.

Findings

Maximum triples in hypergraphs without a $C_6$ in links is $n^{7/3}$

Constructs dense $C_6$-free bipartite graphs with $n^{4/3}$ edges

Extends classical cycle-free graph bounds to hypergraph suspensions

Abstract

For fixed $k\ge 2$, determining the order of magnitude of the number of edges in an $n$-vertex bipartite graph not containing $C_{2k}$, the cycle of length $2k$, is a long-standing open problem. We consider an extension of this problem to triple systems. In particular, we prove that the maximum number of triples in an $n$-vertex triple system which does not contain a $C_6$ in the link of any vertex, has order of magnitude $n^{7/3}$. Additionally, we construct new families of dense $C_6$-free bipartite graphs with $n$ vertices and $n^{4/3}$ edges in order of magnitude.