Large hypergraphs without tight cycles

Barnabás Janzer

doi:10.5070/c61055374

Abstract

An \(r\)-uniform tight cycle of length \(\ell>r\) is a hypergraph with vertices \(v_1,\dots,v_\ell\) and edges \(\{v_i,v_{i+1},\dots,v_{i+r-1}\}\) (for all \(i\)), with the indices taken modulo \(\ell\). It was shown by Sudakov and Tomon that for each fixed \(r\geq 3\), an \(r\)-uniform hypergraph on \(n\) vertices which does not contain a tight cycle of any length has at most \(n^{r-1+o(1)}\) hyperedges, but the best known construction (with the largest number of edges) only gives \(\Omega(n^{r-1})\) edges. In this note we prove that, for each fixed \(r\geq 3\), there are \(r\)-uniform hypergraphs with \(\Omega(n^{r-1}\log n/\log\log n)\) edges which contain no tight cycles, showing that the \(o(1)\) term in the exponent of the upper bound is necessary.Mathematics Subject Classifications: 05C65, 05C38

Highlights

A well-known basic fact about graphs states that a graph on n vertices containing no cycle of any length has at most n − 1 edges, with this upper bound being tight
Given positive integers r 2 and > r, an r-uniform tight cycle of length is a hypergraph with vertices v1, . . . , v and edges {vi, vi+1, . . . , vi+r−1} for i = 1, . . ., with the indices taken modulo
Let fr(n) denote the maximal number of edges that an r-uniform hypergraph on n vertices can have if it has no subgraph isomorphic to a tight cycle of any length

Summary

Title Large hypergraphs without tight cycles

Powered by the California Digital Library University of California combinatorial theory 1 (2021), #12 combinatorial-theory.org. Submitted: Dec 15, 2020; Accepted: Jul 27, 2021; Published: Dec 15, 2021 © The author. Released under the CC BY license (International 4.0)

Introduction

Findings

Sos and independently