22 research outputs found
Large joints in graphs
We show that if G is a graph of sufficiently large order n containing as many
r-cliques as the r-partite Turan graph of order n; then for some C>0 G has more
than Cn^(r-1) (r+1)-cliques sharing a common edge unless G is isomorphic to the
the r-partite Turan graph of order n. This structural result generalizes a
previous result that has been useful in extremal graph theory.Comment: 9 page
Stability for large forbidden subgraphs
We extend the classical stability theorem of Erdos and Simonovits for
forbidden graphs of logarithmic order.Comment: Some polishing. Updated reference
Graphs with many r-cliques have large complete r-partite subgraphs
We prove that for all and c>0, every graph of order n with at least
cn^{r} cliques of order r contains a complete r-partite graph with each part of
size This result implies a concise form of the
Erd\H{o}s-Stone theorem.Comment: Some polishing. Updated reference
Extremal problems for the p-spectral radius of graphs
The -spectral radius of a graph of order is defined for any real
number as
The most remarkable feature of is that it
seamlessly joins several other graph parameters, e.g., is the Lagrangian, is the spectral
radius and is the number of edges. This
paper presents solutions to some extremal problems about , which are common generalizations of corresponding edge and
spectral extremal problems.
Let be the -partite Tur\'{a}n graph of order
Two of the main results in the paper are:
(I) Let and If is a -free graph of order
then unless
(II) Let and If is a graph of order with then has an edge contained in at least
cliques of order where is a positive number depending
only on and Comment: 21 pages. Some minor corrections in v
Ramsey Goodness and Beyond
In a seminal paper from 1983, Burr and Erdos started the systematic study of
Ramsey numbers of cliques vs. large sparse graphs, raising a number of
problems. In this paper we develop a new approach to such Ramsey problems using
a mix of the Szemeredi regularity lemma, embedding of sparse graphs, Turan type
stability, and other structural results. We give exact Ramsey numbers for
various classes of graphs, solving all but one of the Burr-Erdos problems.Comment: A new reference is adde