Work

On r-wise t-intersecting Uniform Families

Year: 2025

Type: article

Abstract: We consider families, $$\mathcal {F}$$ of k-subsets of an n-set. For integers $$r\ge 2$$ , $$t\ge 1$$ , $$\mathcal {F}$$ is called r-wise t-intersecting if any r of its members have at least t elements in common. The most natural construction of such a family is the full t-star, consisting of all k-sets containing a fixed t-set. In the case $$r=2$$ the Exact Erdős-Ko-Rado Theorem shows that the full t-star is largest if $$n\ge (t+1)(k-t+1)$$ . In the present paper, we prove that for $$n\ge (2.5t)^{1/(r-1)}(k-t)+k$$ , the full t-star is largest in case of $$r\ge 3$$ . Examples show that the exponent $$\frac{1}{r-1}$$ is best possible. This represents a considerable improvement on a recent result of Balogh and Linz.

Source: COMBINATORICA

DOI: 10.1007/s00493-025-00166-y

Authors: Peter Frankl, Jian Wang

Institutions: HUN-REN Alfréd Rényi Institute of Mathematics, Taiyuan University of Technology

Language: English

Topic: Limits and Structures in Graph Theory

Subfield: Discrete Mathematics and Combinatorics

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,290

Funder: National Natural Science Foundation of China