An Erd\H{o}s-Ko-Rado theorem for binary codes

TL;DR

This paper investigates intersecting families of binary words, proving that maximum 3-wise intersecting families are stars, and provides a new proof for the known structure of maximum intersecting families over larger alphabets.

Contribution

It establishes that all maximum 3-wise intersecting families of binary words are stars and offers a new proof for the structure of maximum intersecting families over larger alphabets.

Findings

Maximum 3-wise intersecting families in binary are stars.

For alphabets of size ≥ 3, maximum intersecting families are exactly stars.

Provides a new proof for known results on larger alphabets.

Abstract

We study intersecting families of words from the Erd\H{o}s-Ko-Rado perspective. When the alphabet size is $2$, a maximum intersecting family is not necessarily a star. However, we prove that every maximum $3$-wise intersecting family is a star. We also present a new proof of the known result for alphabets of size at least $3$: maximum intersecting families of words are exactly the stars.