# ZarankiewiczDB,

an online database of Zarankiewicz numbers.

ZarankiewiczDB is an online database of numbers, matrices and bounds for the Zarankiewicz problem and related problems.

• The Zarankiewicz problem [Zar51] asks for the maximum number of 1s in an $m{\times}n$ matrix with no $s{\times}t$ minor of only 1s. This maximum is the Zarankiewicz number $z(m,n;~s,t)$. The number of 1s is the weight, and a matrix satisfying this property is $(s,t)$-rectangle-free. As the case $s = t = 2$ is most studied, for brevity we write $z(m,n)$.
• The multicolor Zarankiewicz problem asks for the minimum number $\chi(m,n;~s,t)$ of colors needed to fill in an $m{\times}n$ matrix so that each color class is $(s,t)$-rectangle-free.
• The block Zarankiewicz problem asks for the maximum weight $z(m,n \mid \Sigma)$ of a rectangle-free matrix whose entries are blocks from a set $\Sigma$. The entry sets of most interest are permutation matrices (i.e. a symmetric group), and partial permutation matrices. When $\Sigma$ is the full set of $a{\times}b$ partial permutation matrices, we write $z(m,n \mid a,b)$.

For more information, see theorems & references.