# Large homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs

## TL;DR

This paper investigates the structure of bipartite graphs avoiding certain subgraphs, showing that for strongly acyclic graphs, the largest homogeneous subgraphs grow linearly with the size of the graph, except for four special cases.

## Contribution

It proves that the maximum size of homogeneous subgraphs in bipartite graphs avoiding strongly acyclic subgraphs is linear in the number of vertices, with four exceptions.

## Key findings

- h(Forb(n, H)) is linear in n for all strongly acyclic H except four.
- For non-strongly acyclic H, h(Forb(n, H)) is O(n^{1-s}) for some s>0.
- The paper characterizes the growth of homogeneous subgraphs in bipartite graphs with forbidden induced subgraphs.

## Abstract

For a bipartite graph G, let h(G) be the largest t such that either G or the bipartite complement of G contain K_{t,t}. For a class F of graphs, let h(F)= min {h(G): G\in F}. We say that a bipartite graph H is strongly acyclic if neither H nor its bipartite complement contain a cycle. By Forb(n, H) we denote a set of bipartite graphs with parts of sizes n each, that do not contain H as an induced bipartite subgraph respecting the sides. One can easily show that h(Forb(n,H))= O(n^{1-s}) for a positive s if H is not strongly acyclic. Here, we prove that h(Forb(n, H)) is linear in n for all strongly acyclic graphs except for four graphs.