A note on extremal intersecting linear Ryser systems

TL;DR

This paper investigates extremal properties of intersecting linear Ryser systems, establishing bounds on the minimum number of lines needed for certain transversal numbers and exploring conditions under which Ryser's conjecture holds.

Contribution

It proves a lower bound for the number of lines in intersecting r-partite linear systems with maximum transversal number, and provides conditions ensuring Ryser's conjecture holds for such systems.

Findings

Lower bound for the number of lines in intersecting systems with maximum transversal number.

Exact values of the minimum number of lines for small r.

Conditions on the maximum size of line subsets ensuring Ryser's conjecture validity.

Abstract

A famous conjecture of Ryser states that any $r$-partite set system has transversal number at most $r-1$ times their matching number. This conjecture is only known to be true for $r\leq3$ in general, for $r\leq5$ if the set system is intersecting, and for $r\leq9$ if the intersecting set system is linear. In this note, we deal with Ryser's Conjecture for intersecting $r$-partite linear systems; that is, if $\tau$ is the transversal number for an intersecting $r$-partite linear system, then Ryser's Conjecture states that $\tau\leq r-1$. If this conjecture is true, this is known to be sharp for $r$ for which there exists a projective plane of order $r-1$. There has also been considerable effort to find intersecting $r$-partite set systems whose transversal number is $r-1$. In this note, the following is proved: if $r\geq4$ is an even integer, then $f_l(r)\geq3(r-2)+1$, where $f_l(r)$ is the minimum number of lines of an intersecting $r$-partite linear system whose transversal number is $r-1$. This lower bound gives an exact value for $f_l(r)$, for some small values of $r$. Also, we prove that any $r$-partite linear system satisfies $\tau\leq r-1$ if $\nu_2\leq r$ for all $r\geq3$ odd integer and $\nu_2\leq r-1$ for all $r\geq4$ even integer, where $\nu_2$ is the maximum cardinality of a subset of lines $R\subseteq\mathcal{L}$ such that every triplet of different elements of $R$ does not have a common point.