Clique density vs blowups

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 $\Omega ({n \over {\log n}})$ . 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 <jats:inline-graphic xmlns:xlink="http://www.w3.org/19