[Paper Review] On the Zarankiewicz problem for graphs with bounded VC-dimension

Oliver Janzer, Cosmin Pohoata|arXiv (Cornell University)|Aug 31, 2020

Limits and Structures in Graph Theory19 references4 citations

TL;DR

This paper resolves a key gap in extremal graph theory by showing that for bipartite graphs with VC-dimension at most $d \geq 3$, the number of edges in a $K_{k,k}$-free graph is $o(n^{2-1/d})$, improving upon the previously known $O(n^{2-1/d})$ bound. The result uses a novel application of the hypergraph removal lemma and double counting arguments to rule out dense configurations in bounded VC-dimension settings.

ABSTRACT

The problem of Zarankiewicz asks for the maximum number of edges in a bipartite graph on $n$ vertices which does not contain the complete bipartite graph $K_{k,k}$ as a subgraph. A classical theorem due to Kővári, Sós, and Turán says that this number of edges is $O\left(n^{2 - 1/k} ight)$. An important variant of this problem is the analogous question in bipartite graphs with VC-dimension at most $d$, where $d$ is a fixed integer such that $k \geq d \geq 2$. A remarkable result of Fox, Pach, Sheffer, Suk, and Zahl [J. Eur. Math. Soc. (JEMS), no. 19, 1785-1810] with multiple applications in incidence geometry shows that, under this additional hypothesis, the number of edges in a bipartite graph on $n$ vertices and with no copy of $K_{k,k}$ as a subgraph must be $O\left(n^{2 - 1/d} ight)$. This theorem is sharp when $k=d=2$, because by design any $K_{2,2}$-free graph automatically has VC-dimension at most $2$, and there are well-known examples of such graphs with $Ω\left(n^{3/2} ight)$ edges. However, it turns out this phenomenon no longer carries through for any larger $d$. We show the following improved result: the maximum number of edges in bipartite graphs with no copies of $K_{k,k}$ and VC-dimension at most $d$ is $o(n^{2-1/d})$, for every $k \geq d \geq 3$.

Motivation & Objective

• To close a gap in the Zarankiewicz problem under bounded VC-dimension, particularly for $d \geq 3$, where prior bounds were not known to be tight.
• To show that the $O(n^{2-1/d})$ bound from Fox et al. is not sharp for $d \geq 3$, even though it is tight for $d = 2$.
• To establish that $K_{k,k}$-free bipartite graphs with VC-dimension at most $d \geq 3$ must have significantly fewer edges than $n^{2-1/d}$.
• To investigate the sharpness of the $O(n^{2-1/d})$ bound in the context of VC-dimension and propose a conjecture for a stronger, polynomially smaller exponent.

Proposed method

• Applying the hypergraph removal lemma to analyze configurations of vertices with bounded VC-dimension and shared neighborhoods.
• Using double counting arguments to bound the number of $q$-sets in neighborhoods that could support dense substructures.
• Analyzing the number of common neighbors and disjoint neighborhood sets to derive a contradiction when too many such configurations exist.
• Employing a probabilistic deletion argument to construct lower bounds, based on the ratio $\frac{v(H)-2}{e(H)-1}$ for forbidden subgraphs.
• Using the structure of a specific forbidden graph $F$ with VC-dimension $d+1$ to enforce bounded VC-dimension in the construction.
• Applying a standard amplification and random partitioning technique to derive a bipartite graph with desired sparsity and VC-dimension properties.

Experimental results

Research questions

• RQ1 Is the $O(n^{2-1/d})$ bound for $K_{k,k}$-free bipartite graphs with VC-dimension at most $d$ tight for $d \geq 3$?
• RQ2 Can the exponent in the upper bound be improved beyond $n^{2-1/d}$ under the same VC-dimension constraint?
• RQ3 What is the best possible exponent in the edge bound for $K_{k,k}$-free bipartite graphs with VC-dimension at most $d \geq 3$?
• RQ4 Can the hypergraph removal lemma be used in a way that yields significantly better quantitative bounds for bounded VC-dimension hypergraphs?
• RQ5 How does the VC-dimension constraint affect the extremal edge density when $k > d \geq 3$?

Key findings

• For $k \geq d \geq 3$, the maximum number of edges in a $K_{k,k}$-free bipartite graph with VC-dimension at most $d$ is $o(n^{2-1/d})$, improving the prior $O(n^{2-1/d})$ bound.
• The result is sharp in the sense that the $O(n^{2-1/d})$ bound is tight only for $d = 2$, and fails to be tight for $d \geq 3$.
• The proof relies on a contradiction derived from overcounting $q$-sets with shared neighborhoods, showing that too many such sets would violate the VC-dimension constraint.
• A lower bound construction via the probabilistic method yields $\Omega(n^{2-1/(d-2-C/d)})$ edges for $k \geq 2d-5$, showing the exponent in the upper bound cannot be improved beyond $2-1/(d-2)$ in general.
• The authors conjecture that the true bound is $O(n^{2-1/d-\epsilon})$ for some $\epsilon > 0$, suggesting a significant gap between current upper and lower bounds.
• The result implies that the VC-dimension constraint, while not sufficient to force subquadratic density, does force a nontrivial improvement over the classical Kővári–Sós–Turán bound when $d \geq 3$.