Some remarks on the Zarankiewicz problem

The Zarankiewicz problem asks for an estimate on z(m, n; s, t), the largest number of 1’s in an m × n matrix with all entries 0 or 1 containing no s × t submatrix consisting entirely of 1’s. We show that a classical upper bound for z(m, n; s, t) due to Kővári, Sós and Turán is tight up to the constant for a broad range of parameters. The proof relies on a new quantitative variant of the random algebraic method.

Separating the k-party communication complexity hierarchy: an application of the Zarankiewicz problem

special issue in honor of Laci Babai's 60th birthday: Combinatorics, Groups, Algorithms, and Complexity International audience For every positive integer k, we construct an explicit family of functions f : \0, 1\(n) -\textgreater \0, 1\ which has (k + 1) - party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Omega (n/k(4)2(k)) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our construction relies on known explicit constructions for a famous open problem of K. Zarankiewicz, namely, to find the maximum number of edges in a graph on n vertices that does not contain K-s,K-t as a subgraph.

Spectral Extrema for Graphs: The Zarankiewicz Problem

Let $G$ be a graph on $n$ vertices with spectral radius $\lambda$ (this is the largest eigenvalue of the adjacency matrix of $G$). We show that if $G$ does not contain the complete bipartite graph $K_{t ,s}$ as a subgraph, where $2\le t \le s$, then $$\lambda \le \Big((s-1)^{1/t }+o(1)\Big)n^{1-1/t }$$ for fixed $t$ and $s$ while $n\to\infty$. Asymptotically, this bound matches the Kővári-Turán-Sós upper bound on the average degree of $G$ (the Zarankiewicz problem).