A note on uniform intersecting families with maximum transversal size

TL;DR

This paper constructs specific intersecting families with maximum transversal size, providing new bounds and properties, including a proof that q(4)=9 and a construction for k=2^m-1 with certain transversal size properties.

Contribution

It introduces novel constructions of intersecting families with maximum transversal size and applies these to determine q(4) and to build families with specific size and transversal properties.

Findings

Constructed an intersecting k-family with transversal size ⌈(k+1)/2⌉ and length k+1.

Proved that q(4) = 9.

Built a k-family for k=2^m-1 with length 2k+1 and transversal size ≥ (2k+1)/3.

Abstract

We construct an intersecting $k$-family of transversal size $\lceil \frac{k+1}{2} \rceil$ and length $k+1$ and study some of its properties. We use this family to prove that $q(4) = 9$. We also construct a $k$-family for $k = 2^m - 1$ of length $2k+1$ and transversal size at least $(2k+1)/3$.