A bipartite version of the Erd\H{o}s $-$ McKay conjecture

TL;DR

This paper proves a bipartite analogue of a longstanding conjecture, showing that bipartite graphs with small balanced homogeneous sets contain all intermediate-sized induced subgraphs.

Contribution

It introduces a bipartite version of the Erdős-McKay conjecture and proves it for graphs with small balanced homogeneous sets.

Findings

Bipartite graphs with small balanced homogeneous sets contain all sizes of induced subgraphs.

The bipartite analogue of the Erdős-McKay conjecture is validated.

The result extends the understanding of structure in bipartite graphs.

Abstract

An old conjecture of Erd\H{o}s and McKay states that if all homogeneous sets in an $n$-vertex graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega (n^2)\}$. We prove a bipartite analogue of the conjecture: if all balanced homogeneous sets in an $n \times n$ bipartite graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega (n^2)\}$.