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On cliques in hypergraphs

Jun Gao

TL;DR

This paper addresses the maximum number of distinct clique sizes, $g(n,k)$, in a $k$-uniform hypergraph on $n$ vertices, where a clique is a maximal complete subgraph. The main result shows that for every $k \ge 3$ and every $C \ge 0$ there exists $N=N(k,C)$ such that $n \ge N$ forces $g(n,k) \le n-C$, i.e., at most $n-C$ clique sizes, answering Erdős's question in the affirmative for all $k \ge 3$. The $k=3$ case is proved first by partitioning maximal cliques and applying a clique-union fact to bound the number of cliques, and the general case proceeds by constructing a $(k-1,C)$-layered tree from a chain of maximal cliques and applying a bound on the size of such trees. Consequently, the unbounded growth of $f(n,k)=n-g(n,k)$ is established, with a precise bound for $k=3$ involving iterated logarithms.

Abstract

We prove that for any $k \ge 3$, every $k$-uniform hypergraph on $n$ vertices contains at most $n - ω(1)$ different sizes of cliques (maximal complete subgraphs). In particular, the 3-uniform case answers a question of Erdős.

On cliques in hypergraphs

TL;DR

This paper addresses the maximum number of distinct clique sizes, , in a -uniform hypergraph on vertices, where a clique is a maximal complete subgraph. The main result shows that for every and every there exists such that forces , i.e., at most clique sizes, answering Erdős's question in the affirmative for all . The case is proved first by partitioning maximal cliques and applying a clique-union fact to bound the number of cliques, and the general case proceeds by constructing a -layered tree from a chain of maximal cliques and applying a bound on the size of such trees. Consequently, the unbounded growth of is established, with a precise bound for involving iterated logarithms.

Abstract

We prove that for any , every -uniform hypergraph on vertices contains at most different sizes of cliques (maximal complete subgraphs). In particular, the 3-uniform case answers a question of Erdős.
Paper Structure (3 sections, 3 theorems, 14 equations, 1 figure)

This paper contains 3 sections, 3 theorems, 14 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['thm: main']}
  3. Concluding remarks

Key Result

Theorem 1.1

For any integers $k \ge 3$ and $C\ge 0$, there exists a constant $N=N(k,C)$ such that for all $n\ge N$, every $n$-vertex $k$-uniform hypergraph contains no more than $n-C$ different sizes of cliques.

Figures (1)

  • Figure 1: Properties of the path in $T$.

Theorems & Definitions (13)

  • Theorem 1.1
  • Definition 2.1: $(k,C)$-layered tree
  • Lemma 2.2
  • proof
  • Claim 2.3
  • proof : Proof of the claim
  • Claim 2.4
  • proof : Proof of the claim
  • proof : Proof of Theorem \ref{['thm: main']}
  • Claim 2.6
  • ...and 3 more