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Let be a graph of density p on n vertices. Following Erdős, Łuczak, and Spencer, an m‐vertex subgraph H of G is called full if H has minimum degree at least . Let denote the order of a largest full subgraph of G. If is a nonnegative integer, define
Erdős, Łuczak, and Spencer proved that for ,
In this article, we prove the following lower bound: for ,
Furthermore,...
A long-standing conjecture of Erdős and Simonovits is that ex(n,C2k), the maximum number of edges in an n-vertex graph without a 2k-gon is asymptotically 12n1+1/k as n tends to infinity. This was known almost 40 years ago in the case of quadrilaterals. In this paper, we construct a counterexample to the conjecture in the case of hexagons. For infinitely many n, we prove thatex(n,C6)>3(5-2)(5-1)4/3n4/3+O(n)>0...
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