5 research outputs found
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
Some remarks on the Zarankiewicz problem
The Zarankiewicz problem asks for an estimate on , the largest
number of 's in an matrix with all entries or
containing no submatrix consisting entirely of 's. We show that
a classical upper bound for due to K\H{o}v\'ari, S\'os and
Tur\'an 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.Comment: 6 page
Shatter functions with polynomial growth rates
International audienceWe study how a single value of the shatter function of a set system restricts its asymptotic growth. Along the way, we refute a conjecture of Bondy and Hajnal which generalizes Sauer's Lemma