Hypergraph removal with polynomial bounds

Abstract

Given a fixed $k$-uniform hypergraph $F$, the $F$-removal lemma states that every hypergraph with few copies of $F$ can be made $F$-free by the removal of few edges. Unfortunately, for general $F$, the constants involved are given by incredibly fast-growing Ackermann-type functions. It is thus natural to ask for which $F$ one can prove removal lemmas with polynomial bounds. One trivial case where such bounds can be obtained is when $F$ is $k$-partite. Alon proved that when $k=2$ (i.e. when dealing with graphs), only bipartite graphs have a polynomial removal lemma. Kohayakawa, Nagle and R\"odl conjectured in 2002 that Alon's result can be extended to all $k>2$, namely, that the only $k$-graphs $F$ for which the hypergraph removal lemma has polynomial bounds are the trivial cases when $F$ is $k$-partite. In this paper we prove this conjecture.