Continuous Tur\'an numbers

TL;DR

This paper introduces a continuous extremal function for subsets of b2, connecting it to classical extremal problems like Zarankiewicz and matrix extremal functions, and extends results to infinite and rational subsets.

Contribution

It defines a new continuous extremal function px(n, P) for subsets of b2, establishing its relation to matrix extremal functions and extending classical theorems to infinite and rational point sets.

Findings

px(n, P) is b2-equivalent to the matrix extremal function ex(n, M)

For open subsets P, px(n, P) = b2 growth is quadratic, b2n^2

Extended Kf6ve1ri-Sf3s-Ture1n theorem to certain infinite subsets

Abstract

In this paper, we define a notion of containment and avoidance for subsets of $\mathbb{R}^2$. Then we introduce a new, continuous and super-additive extremal function for subsets $P \subseteq \mathbb{R}^2$ called $px(n, P)$, which is the supremum of $\mu_2(S)$ over all open $P$-free subsets $S \subseteq [0, n]^2$, where $\mu_2(S)$ denotes the Lebesgue measure of $S$ in $\mathbb{R}^2$. We show that $px(n, P)$ fully encompasses the Zarankiewicz problem and more generally the 0-1 matrix extremal function $ex(n, M)$ up to a constant factor. More specifically, we define a natural correspondence between finite subsets $P \subseteq \mathbb{R}^2$ and 0-1 matrices $M_P$, and we prove that $px(n, P) = \Theta(ex(n, M_P))$ for all finite subsets $P \subseteq \mathbb{R}^2$, 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 $px(n, P)$ grows faster than $ex(n, M)$ for all fixed 0-1 matrices $M$. In particular, we show that $px(n, P) = \Theta(n^{2})$ for any open subset $P \subseteq \mathbb{R}^2$. We prove an even stronger result, that if $Q_P$ is the set of points with rational coordinates in any open subset $P \subseteq \mathbb{R}^2$, then $px(n, Q_P) = \Theta(n^2)$. Finally, we obtain a strengthening of the K\H{o}vari-S\'{o}s-Tur\'{a}n theorem that applies to infinite subsets of $\mathbb{R}^2$. Specifically, for subsets $P_{s, t, c} \subseteq \mathbb{R}^2$ 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 $px(n, P_{s, t,c}) = O(s^{\frac{1}{t}}n^{2-\frac{1}{t}})$, 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 $c$.