 
Summary: Bipartite subgraphs
(Final Version; appeared in Combinatorica 16 (1996), 301311.)
Noga Alon
Abstract
It is shown that there exists a positive c so that for any large integer m, any graph with 2m2
edges contains a bipartite subgraph with at least m2
+ m/2 + c
m edges. This is tight up to
the constant c and settles a problem of Erdos. It is also proved that any trianglefree graph with
e > 1 edges contains a bipartite subgraph with at least e
2 +c e4/5
edges for some absolute positive
constant c . This is tight up to the constant c .
1 Introduction
For a graph G, let f(G) denote the maximum number of edges in a bipartite subgraph of G. For
a positive integer e let f(e) denote the minimum value of f(G), as G ranges over all graphs with
e edges. Thus, f(e) is the largest integer f such that any graph with e edges contains a bipartite
subgraph with at least f(e) edges. Edwards [4], [5] proved that for every e
f(e)
