Covers in partitioned intersecting hypergraphs

Abstract

Given an integer $r$ and a vector $\overrightarrow{a}=(a_1,\dots ,a_p)$ of positive numbers with ${\sum }_{i⩽p}{a}_i=r$, an $r$-uniform hypergraph $H$ is said to be $\overrightarrow{a}$-partitioned if $V\left(H\right)={\bigcup }_{i⩽p}{V}_i$, where the sets $V_i$ are disjoint, and $|e\cap V_i|=a_i$ for all $e\in H,i⩽p$. A $\overrightarrow{1}$-partitioned hypergraph is said to be $r$-partite. Let $t\left(\overrightarrow{a}\right)$ be the maximum, over all intersecting $\overrightarrow{a}$-partitioned hypergraphs $H$, of the minimal size of a cover of $H$. A famous conjecture of Ryser is that $t\left(\overrightarrow{1}\right)⩽r-1$. Tuza (1983) conjectured that if $r>2$ then $t\left(\overrightarrow{a}\right)=r$ for every two components vector $\overrightarrow{a}=(a,b)$. We prove this conjecture whenever $a\ne b$, and also for $\overrightarrow{a}=(2,2)$ and $\overrightarrow{a}=(4,4)$.