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Zarankiewicz's problem via $ε$-t-nets

Chaya Keller, Shakhar Smorodinsky

Abstract

The classical Zarankiewicz's problem asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{t,t}$. In one of the cornerstones of extremal graph theory, Kővári Sós and Turán proved an upper bound of $O(n^{2-\frac{1}{t}})$. In a celebrated result, Fox et al. obtained an improved bound of $O(n^{2-\frac{1}{d}})$ for graphs of VC-dimension $d$ (where $d<t$). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. At SODA'23, Chan and Har-Peled further improved Basit et al.'s bounds and presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of $O(n \log \log n)$ for the incidence graph of points and pseudo-discs in the plane. In this paper we present a new approach to Zarankiewicz's problem, via $ε$-t-nets - a recently introduced generalization of the classical notion of $ε$-nets. We show that the existence of `small'-sized $ε$-t-nets implies upper bounds for Zarankiewicz's problem. Using the new approach, we obtain a sharp bound of $O(n)$ for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain a short proof of the $O(n^{2-\frac{1}{d}})$ bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp $O(n\frac{\log n}{\log \log n})$ bound for the intersection graph of two families of axis-parallel rectangles.

Zarankiewicz's problem via $ε$-t-nets

Abstract

The classical Zarankiewicz's problem asks for the maximum number of edges in a bipartite graph on vertices which does not contain the complete bipartite graph . In one of the cornerstones of extremal graph theory, Kővári Sós and Turán proved an upper bound of . In a celebrated result, Fox et al. obtained an improved bound of for graphs of VC-dimension (where ). Basit, Chernikov, Starchenko, Tao and Tran improved the bound for the case of semilinear graphs. At SODA'23, Chan and Har-Peled further improved Basit et al.'s bounds and presented (quasi-)linear upper bounds for several classes of geometrically-defined incidence graphs, including a bound of for the incidence graph of points and pseudo-discs in the plane. In this paper we present a new approach to Zarankiewicz's problem, via -t-nets - a recently introduced generalization of the classical notion of -nets. We show that the existence of `small'-sized -t-nets implies upper bounds for Zarankiewicz's problem. Using the new approach, we obtain a sharp bound of for the intersection graph of two families of pseudo-discs, thus both improving and generalizing the result of Chan and Har-Peled from incidence graphs to intersection graphs. We also obtain a short proof of the bound of Fox et al., and show improved bounds for several other classes of geometric intersection graphs, including a sharp bound for the intersection graph of two families of axis-parallel rectangles.
Paper Structure (13 sections, 13 theorems, 22 equations, 3 figures, 1 algorithm)

This paper contains 13 sections, 13 theorems, 22 equations, 3 figures, 1 algorithm.

Table of Contents

  1. Introduction
  2. Our results
  3. Follow-up work
  4. Organization of the paper
  5. From $\epsilon$-$t$-Nets to $K_{t,t}$-free Bipartite Graphs
  6. A recursive upper bound on the size of $K_{t,t}$-free bipartite graphs
  7. Graphs with bounded VC-dimension
  8. $K_{t,t}$-free bipartite intersection graphs of pseudo-discs
  9. $K_{t,t}$-free Intersection Graph of Families of Non-Piercing Regions
  10. $K_{t,t}$-free Bipartite Intersection Graphs of Axis-parallel Rectangles
  11. Proof of Theorem \ref{['thm:rect-intro']}
  12. $K_{t,t}$-free incidence graph of points and pseudo-discs
  13. Comparison between Theorem \ref{['thm:main-boundedvc']} and Theorem \ref{['thm:VCour']}

Key Result

Theorem 1.2

Let $t \geq 2$ and let $G_{A,B}$ be a bipartite graph with $|A|=m$ and $|B|=n$, satisfying $\pi_G(\ell)=O(\ell^d)$ and $\pi_G^*(\ell)=O(\ell^{d^*})$ for all $\ell$. If $G$ is $K_{t,t}$-free, then

Figures (3)

  • Figure 1: The planar drawing of $v=\{h_i,h_j\}$ (in bold).
  • Figure 2: In this figure, the planar drawing of $\{h_i,h_j\}$ intersects the planar drawing of $\{h_i',h_j'\}$. However, $v'$ intersects three horizontal segments, and therefore $\{h_i',h_j'\} \notin E(Del(J))$.
  • Figure 3: An illustration for the discussion in Appendix \ref{['App:Fox']} in the case $m=n, d=d^*$. The blue graph represents the method of FPSSZ17, and the red one represents the method suggested in this work.

Theorems & Definitions (29)

  • Theorem 1.2: FPSSZ17
  • Definition 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Theorem 1.7
  • Theorem 2.1
  • proof
  • proof : Proof of Theorem \ref{['thm:VCour']}
  • Definition 2.2
  • ...and 19 more