180 research outputs found
Erd\H{o}s-Ko-Rado for random hypergraphs: asymptotics and stability
We investigate the asymptotic version of the Erd\H{o}s-Ko-Rado theorem for
the random -uniform hypergraph . For , let and . We show that with probability
tending to 1 as , the largest intersecting subhypergraph of
has size , for any . This lower bound on is
asymptotically best possible for . For this range of and ,
we are able to show stability as well.
A different behavior occurs when . In this case, the lower bound on
is almost optimal. Further, for the small interval , the largest intersecting subhypergraph of
has size , provided that .
Together with previous work of Balogh, Bohman and Mubayi, these results
settle the asymptotic size of the largest intersecting family in
, for essentially all values of and
Balanced supersaturation for some degenerate hypergraphs
A classical theorem of Simonovits from the 1980s asserts that every graph
satisfying must contain copies of . Recently, Morris and
Saxton established a balanced version of Simonovits' theorem, showing that such
has copies of , which
are `uniformly distributed' over the edges of . Moreover, they used this
result to obtain a sharp bound on the number of -free graphs via the
container method. In this paper, we generalise Morris-Saxton's results for even
cycles to -graphs. We also prove analogous results for complete
-partite -graphs.Comment: Changed title, abstract and introduction were rewritte
The history of degenerate (bipartite) extremal graph problems
This paper is a survey on Extremal Graph Theory, primarily focusing on the
case when one of the excluded graphs is bipartite. On one hand we give an
introduction to this field and also describe many important results, methods,
problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version
of our survey presented in Erdos 100. In this version 2 only a citation was
complete
Supersaturation for hereditary properties
Let be a collection of -uniform hypergraphs, and let . It is known that there exists such that the
probability of a random -graph in not containing an induced
subgraph from is . Let each graph in
have at least vertices. We show that in fact for every
, there exists
such that the probability of a random -graph in containing less
than induced subgraphs each lying in is at most
.
This statement is an analogue for hereditary properties of the
supersaturation theorem of Erd\H{o}s and Simonovits. In our applications we
answer a question of Bollob\'as and Nikiforov.Comment: 5 pages, submitted to European Journal of Combinatoric
Color the cycles
The cycles of length k in a complete graph on n vertices are colored in such a way that edge-disjoint cycles get distinct colors. The minimum number of colors is asymptotically determined. © 2013
The Turán problem for hypergraphs of fixed size
We obtain a general bound on the Turán density of a hypergraph in terms of the number of edges that it contains. If F is an r-uniform hypergraph with f edges
we show that [pi](F) =3 and f->[infinity]
- …