15 research outputs found
On the KŁR conjecture in random graphs
The KŁR conjecture of Kohayakawa, Łuczak, and Rödl is a statement that allows one to prove that asymptotically almost surely all subgraphs of the random graph Gn,p, for sufficiently large p := p(n), satisfy an embedding lemma which complements the sparse regularity lemma of Kohayakawa and Rödl. We prove a variant of this conjecture which is sufficient for most known applications to random graphs. In particular, our result implies a number of recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical extremal combinatorial theorems. We also discuss several further applications.</p
Radial variation of analytic functions with non-tangential boundary limits almost everywhere
On the KLR conjecture in random graphs
The K{\L}R conjecture of Kohayakawa, {\L}uczak, and R\"odl is a statement
that allows one to prove that asymptotically almost surely all subgraphs of the
random graph G_{n,p}, for sufficiently large p : = p(n), satisfy an embedding
lemma which complements the sparse regularity lemma of Kohayakawa and R\"odl.
We prove a variant of this conjecture which is sufficient for most known
applications to random graphs. In particular, our result implies a number of
recent probabilistic versions, due to Conlon, Gowers, and Schacht, of classical
extremal combinatorial theorems. We also discuss several further applications.Comment: 33 page