Some remarks on the Zarankiewicz problem

Abstract

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.

Extremal graphs without exponentially-small bicliques

“…Part (b) is an improvement on the result of Conlon [7], who proved the Ω s (mn 1−1/s ) bound under the condition log n m ≤ c ′ s t 1/(s−1) . In the same article, Conlon asked if the bound holds for log n m ≤ t/s, which would be tight if true.…”

“…Since the polynomial g is independent of L 0 and R 0 , it follows, by Lemma 4(b) applied with Y = L × R to the single random polynomial g, that the edge set E(G) = (L × R) ∩ V(g) is of size at least |L||R|/2q with probability at least 1 − O(q −2s+1 ). In view of (7), it then follows that…”