Large subgraphs without complete bipartite graphs

TL;DR

This paper investigates the guaranteed size of the largest subgraph without a complete bipartite graph $K_{r,s}$ in any graph with a given number of edges, extending the discussion to hypergraphs.

Contribution

It provides bounds on the size of the largest $K_{r,s}$-free subgraph in graphs with $m$ edges and explores the analogous problem for hypergraphs.

Findings

Bounds on the size of $K_{r,s}$-free subgraphs in graphs with $m$ edges

Extension of results to hypergraph settings

Answers a specific open question in extremal graph theory

Abstract

In this note, we answer the following question of Foucaud, Krivelevich and Perarnau. What is the size of the largest $K_{r,s}$-free subgraph one can guarantee in every graph $G$ with $m$ edges? We also discuss the analogous problem for hypergraphs.