Induced C4-free subgraphs with large average degree

Abstract

We prove that there exists a constant C so that, for all $s,k\in N$, if G has average degree at least $k^{C{s}^3}$ and does not contain $K_{s,s}$ as a subgraph then it contains an induced subgraph which is $C_4$-free and has average degree at least k. It was known that some function of s and k suffices, but this is the first explicit bound. We give several applications of this result, including short and streamlined proofs of the following two corollaries.

We show that there exists a constant C so that, for all $s,k\in N$, if G has average degree at least $k^{C{s}^3}$ and does not contain $K_{s,s}$ as a subgraph then it contains an induced subdivision of $K_k$. This is the first quantitative improvement on a well-known theorem of Kühn and Osthus; their proof gives a bound that is triply exponential in both k and s.

We also show that for any hereditary degree-bounded class $F$, there exists a constant $C=C_F$ so that $C^{s^3}$ is a degree-bounding function for $F$. This is the first bound of any type on the rate of growth of such functions.