The multiplication table problem for bipartite graphs

TL;DR

This paper extends Erd ext{"o}s's multiplication table problem to bipartite graphs, establishing a lower bound on the number of distinct induced subgraph sizes based on the number of edges.

Contribution

It generalizes the multiplication table problem to bipartite graphs and provides a lower bound on the number of distinct induced subgraph sizes.

Findings

Set of induced subgraph sizes contains at least (m/(4 4 m)^{12}) elements.

Establishes a quantitative link between edges and induced subgraph size diversity.

Extends classical number theory problem to graph theory context.

Abstract

We investigate the following generalisation of the 'multiplication table problem' of Erd\H{o}s: given a bipartite graph with $m$ edges, how large is the set of sizes of its induced subgraphs? Erd\H{o}s's problem of estimating the number of distinct products $ab$ with $a,b \le n$ is precisely the problem under consideration when the graph in question is the complete bipartite graph $K_{n,n}$. In this note, we prove that the set of sizes of the induced subgraphs of any bipartite graph with $m$ edges contains $\Omega(m/(\log m)^{12})$ distinct elements.