A generalization of Erd\H{o}s--Ko--Rado theorem to $t$-designs in certain semilattices

TL;DR

This paper extends the Erdős–Ko–Rado theorem to t-designs within specific semilattices, providing new intersection theorems applicable to various combinatorial schemes and structures.

Contribution

It generalizes the Erdős–Ko–Rado theorem to t-designs in semilattices, broadening its applicability to multiple combinatorial schemes.

Findings

Established intersection theorems for Hamming and Johnson schemes

Extended results to bilinear forms and Grassmann schemes

Applied to signed sets and partial permutations

Abstract

The Erd\H{o}s--Ko--Rado theorem is extended to designs in semilattices with certain conditions. As an application, we show the intersection theorems for the Hamming schemes, the Johnson schemes, bilinear forms schemes, Grassmann schemes, signed sets, partial permutations and restricted signed sets.