Georgia Institute of Technology College of Sciences

The regularity method for graphs has been well studied for dense graphs, i.e., graphs on $n$ vertices with $\Omega(n^2)$ edges. However, applying it to sparse graphs, i.e., those with $o(n^2)$ edges seems to be a harder problem. In the mid 2010s, the regularity method was extended to dense subgraphs of random graphs thus resolving the KŁR conjecture. Later, in another direction, Conlon, Fox, Sudakov and Zhao proved a removal lemma for $C_5$ in graphs that do not contain any $C_4$ (such graphs on $n$ vertices can contain at most $n^{3/2}$ edges). In this talk, we will consider a similar problem for sparse $3$-uniform hypergraphs. In particular, we consider an application of the regularity method to $3$-uniform hypergraphs whose vertices do not contain $C_4$ in their links and satisfy an additional boundedness condition. This is joint work with Vojtěch Rödl and Mathias Schacht.