Continuous Turán numbers

Abstract

One of the most famous open problems in combinatorics is the Zarankiewicz problem, which asks for the maximum number of ones in an n × n matrix that has no s × t submatrix of all ones. The Kővari-Sós-Turán theorem provides an upper bound of O(s) for this problem for fixed t ≥ 2, which is known to be sharp in some cases. The Zarankiewicz problem is a subproblem of the more general problem of determining the maximum number of ones ex(n, M ) in an n × n 0-1 matrix that avoids the forbidden 0-1 matrix M .In this paper, we define a notion of containment and avoidance for subsets of R 2 . Then we introduce a new, continuous and super-additive extremal function for subsets P ⊆ R 2 called opx(n, P ), which is the supremum of µ 2 (S) over all open P -free subsets S ⊆ [0, n] 2 , where µ 2 (S) denotes the Lebesgue measure of S in R 2 . We show that opx(n, P ) fully encompasses ex(n, M ) up to a constant factor. More specifically, we define a natural correspondence between finite subsets P ⊆ R 2 and 0-1 matrices M P , and we prove that opx(n, P ) = Θ(ex(n, M P )) for all finite subsets P ⊆ R 2 , where the constants in the bounds depend only on the distances between the points in P .We also discuss bounded infinite subsets P for which opx(n, P ) grows faster than ex(n, M ) for all fixed 0-1 matrices M . In particular, we show that opx(n, P ) = Θ(n 2 ) for any open subset P ⊆ R 2 . We prove an even stronger result, that if Q P is the set of points with rational coordinates in any open subset P ⊆ R 2 , then opx(n, Q P ) = Θ(n 2 ). Finally, we obtain a strengthening of the Kővari-Sós-Turán theorem that applies to infinite subsets of R 2 . Specifically, for subsets P s,t,c ⊆ R 2 consisting of t horizontal line segments of length s with left endpoints on the same vertical line with consecutive segments a distance of c apart, we prove that opx(n, P s,t,c ) = O(s, where the constant in the bound depends on t and c. When t = 2, we show that this bound is sharp up to a constant factor that depends on c. We also extend opx(n, P ) to any number of dimensions, and we generalize most of our results including the strengthening of the Kővari-Sós-Turán theorem to any number of dimensions.