5 research outputs found
Tur\'annical hypergraphs
This paper is motivated by the question of how global and dense restriction
sets in results from extremal combinatorics can be replaced by less global and
sparser ones. The result we consider here as an example is Turan's theorem,
which deals with graphs G=([n],E) such that no member of the restriction set
consisting of all r-tuples on [n] induces a copy of K_r.
Firstly, we examine what happens when this restriction set is replaced just
by all r-tuples touching a given m-element set. That is, we determine the
maximal number of edges in an n-vertex such that no K_r hits a given vertex
set.
Secondly, we consider sparse random restriction sets. An r-uniform hypergraph
R on vertex set [n] is called Turannical (respectively epsilon-Turannical), if
for any graph G on [n] with more edges than the Turan number ex(n,K_r)
(respectively (1+\eps)ex(n,K_r), no hyperedge of R induces a copy of K_r in G.
We determine the thresholds for random r-uniform hypergraphs to be Turannical
and to epsilon-Turannical.
Thirdly, we transfer this result to sparse random graphs, using techniques
recently developed by Schacht [Extremal results for random discrete structures]
to prove the Kohayakawa-Luczak-Rodl Conjecture on Turan's theorem in random
graphs.Comment: 33 pages, minor improvements thanks to two referee
On extremal subgraphs of random graphs
Let K-l denote the complete graph on vertices. We prove that there is a constant c = c(l) > 0, such that whenever p >= n(-c), with probability tending to 1 when n goes to infinity, every maximum K-l-free subgraph of the binomial random graph G(n,p) is (l-1)-partite. This answers a question of Babai, Simonovits and Spencer [3]. The proof is based on a tool of independent interest: we show, for instance, that the maximum cut of almost all graphs with M edges, where M >> n, is nearly unique. More precisely, given a maximum cut C of G(n,m), we can obtain all maximum cuts by moving at most O (root n(3/)M) vertices between the parts of C