Number of cliques in graphs with a forbidden subdivision

TL;DR

This paper establishes upper bounds on the number of cliques in graphs that exclude a certain subdivision, showing they grow exponentially with the size of the forbidden subdivision and linearly with the number of vertices.

Contribution

It proves tight exponential bounds on the number of cliques in graphs with no $K_t$-subdivision, answering a longstanding open question.

Findings

Graphs with no $K_t$-subdivision have at most $2^{50t}n$ cliques.

Asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques.

The results confirm exponential bounds in $t$ for the number of cliques.

Abstract

We prove that for all positive integers $t$, every $n$-vertex graph with no $K_t$-subdivision has at most $2^{50t}n$ cliques. We also prove that asymptotically, such graphs contain at most $2^{(5+o(1))t}n$ cliques, where $o(1)$ tends to zero as $t$ tends to infinity. This strongly answers a question of D. Wood asking if the number of cliques in $n$-vertex graphs with no $K_t$-minor is at most $2^{ct}n$ for some constant $c$.