A Step Towards Yuzvinsky's Conjecture

An intercalate matrix $M$ of type $[r,s,n]$ is an $r\times s$ matrix with entries in $\{1,2,\dotsc,n\}$ such that all entries in each row are distinct, all entries in each column are distinct, and all $2 \times 2$ submatrices of $M$ have either $2$ or $4$ distinct entries. Yuzvinsky's Conjecture on intercalate matrices claims that the smallest $n$ for which there is an intercalate matrix of type $[r,s,n]$ is the Hopf-Stiefel function $r \circ s$. In this paper we prove that Yuzvinsky's Conjecture is asimptotically true for $\frac{5}{6}$ of integer pairs $(r,s)$. We prove the Conjecture for $r\le 8$, and we study it in the range $r,s\le 32$.