Clique density vs blowups

TL;DR

This paper investigates how positive clique densities in graphs imply the existence of large blowups or bicliques, extending classical results and providing new bounds and constructions in ordered and incomparability graphs.

Contribution

It generalizes Nikiforov's theorem to ordered and incomparability graphs, establishing new bounds on blowup sizes and connecting the problem to Ramsey theory and regularity lemmas.

Findings

Positive clique density implies large blowups in incomparability graphs.

The function g(k) grows quadratically, with bounds up to a factor of 2.

Blowup sizes of n/g are optimal up to constants.

Abstract

A well-known theorem of Nikiforov asserts that any graph with a positive $K_{r}$-density contains a logarithmic blowup of $K_r$. In this paper, we explore variants of Nikiforov's result in the following form. Given $r,t\in\mathbb{N}$, when a positive $K_{r}$-density implies the existence of a significantly larger (with almost linear size) blowup of $K_t$? Our results include: For an $n$-vertex ordered graph $G$ with no induced monotone path $P_{6}$, if its complement $\overline{G}$ has positive triangle density, then $\overline{G}$ contains a biclique of size $\Omega(\frac{n}{\log{n}})$. This strengthens a recent result of Pach and Tomon. For general $k$, let $g(k)$ be the minimum $r\in \mathbb{N}$ such that for any $n$-vertex ordered graph $G$ with no induced monotone $P_{2k}$, if $\overline{G}$ has positive $K_r$-density, then $\overline{G}$ contains a biclique of size $\Omega(\frac{n}{\log{n}})$. Using concentration of measure and the isodiametric inequality on high dimensional spheres, we provide constructions showing that, surprisingly, $g(k)$ grows quadratically. On the other hand, we relate the problem of upper bounding $g(k)$ to a certain Ramsey problem and determine $g(k)$ up to a factor of 2. Any incomparability graph with positive $K_{r}$-density contains a blowup of $K_r$ of size $\Omega(\frac{n}{\log{n}}).$ This confirms a conjecture of Tomon in a stronger form. In doing so, we obtain a strong regularity type lemma for incomparability graphs with no large blowups of a clique, which is of independent interest. We also prove that any $r$-comparability graph with positive $K_{(2h-2)^{r}+1}$-density contains a blowup of $K_h$ of size $\Omega(n)$, where the constant $(2h-2)^{r}+1$ is optimal. The $\frac{n}{\log n}$ size of the blowups in all our results are optimal up to a constant factor.