Many cliques in bounded-degree hypergraphs

TL;DR

This paper extends the study of maximum clique counts to hypergraphs with bounded degrees, establishing bounds and characterizing extremal structures, including connections to Steiner systems.

Contribution

It generalizes clique bounds from graphs to hypergraphs with degree constraints, providing tight bounds and characterizing extremal hypergraphs.

Findings

Bounds on the number of t-cliques in s-graphs with degree constraints.

Extremal hypergraphs are shadows of Steiner or partial Steiner systems.

Extension of Kruskal-Katona uniqueness results to hypergraph cliques.

Abstract

Recently Chase determined the maximum possible number of cliques of size $t$ in a graph on $n$ vertices with given maximum degree. Soon afterward, Chakraborti and Chen answered the version of this question in which we ask that the graph have $m$ edges and fixed maximum degree (without imposing any constraint on the number of vertices). In this paper we address these problems on hypergraphs. For $s$-graphs with $s\ge 3$ a number of issues arise that do not appear in the graph case. For instance, for general $s$-graphs we can assign degrees to any $i$-subset of the vertex set with $1\le i\le s-1$. We establish bounds on the number of $t$-cliques in an $s$-graph $\mathcal{H}$ with $i$-degree bounded by $\Delta$ in three contexts: $\mathcal{H}$ has $n$ vertices; $\mathcal{H}$ has $m$ (hyper)edges; and (generalizing the previous case) $\mathcal{H}$ has a fixed number $p$ of $u$-cliques for some $u$ with $s\le u \le t$. When $\Delta$ is of a special form we characterize the extremal $s$-graphs and prove that the bounds are tight. These extremal examples are the shadows of either Steiner systems or partial Steiner systems. On the way to proving our uniqueness results, we extend results of F\"uredi and Griggs on uniqueness in Kruskal-Katona from the shadow case to the clique case.