Given an integer $r$ and a vector $\vec{ a}=(a_1, \ldots ,a_p)$ of positive numbers with $\sum_{ i \le p} a_i=r$, an $r$-uniform hypergraph $H$ is said to be $\vec{ a}$-partitioned if $V(H)=\bigcup_{ i \le p}V_i$, where the sets $V_i$ are disjoint, and $|e \cap V_i|=a_i$ for all $e \in H,~~i \le p$. A $\vec{ 1}$-partitioned hypergraph is said to be $r$-partite. Let $t(\vec{ a})$ be the maximum, over all intersecting $\vec{ a}$-partitioned hypergraphs $H$, of the minimal size of a cover of $H$. A famous conjecture of Ryser is that $t(\vec{ 1})\le r-1$. Tuza conjectured that if $r>2$ then $t(\vec{ a})=r$ for every two components vector $\vec{ a}=(a,b)$. We prove this conjecture whenever $a\neq b$, and also for $\vec{ a}=(2,2)$ and $\vec{ a}=(4,4)$.
Categories: math.CO