234 research outputs found
Density version of the Ramsey problem and the directed Ramsey problem
We discuss a variant of the Ramsey and the directed Ramsey problem. First,
consider a complete graph on vertices and a two-coloring of the edges such
that every edge is colored with at least one color and the number of bicolored
edges is given. The aim is to find the maximal size of a
monochromatic clique which is guaranteed by such a coloring. Analogously, in
the second problem we consider semicomplete digraph on vertices such that
the number of bi-oriented edges is given. The aim is to bound the
size of the maximal transitive subtournament that is guaranteed by such a
digraph.
Applying probabilistic and analytic tools and constructive methods we show
that if , (), then where only depend on , while if then . The latter case is
strongly connected to Tur\'an-type extremal graph theory.Comment: 17 pages. Further lower bound added in case $|E_{RB}|=|E_{bi}| =
p{n\choose 2}
Large unavoidable subtournaments
Let denote the tournament on vertices consisting of three disjoint
vertex classes and of size , each of which is oriented as a
transitive subtournament, and with edges directed from to , from
to and from to . Fox and Sudakov proved that given a
natural number and there is such that
every tournament of order which is -far from
being transitive contains as a subtournament. Their proof showed that
and they conjectured that
this could be reduced to . Here we
prove this conjecture.Comment: 9 page
On the number of 4-cycles in a tournament
If is an -vertex tournament with a given number of -cycles, what
can be said about the number of its -cycles? The most interesting range of
this problem is where is assumed to have cyclic triples for
some and we seek to minimize the number of -cycles. We conjecture that
the (asymptotic) minimizing is a random blow-up of a constant-sized
transitive tournament. Using the method of flag algebras, we derive a lower
bound that almost matches the conjectured value. We are able to answer the
easier problem of maximizing the number of -cycles. These questions can be
equivalently stated in terms of transitive subtournaments. Namely, given the
number of transitive triples in , how many transitive quadruples can it
have? As far as we know, this is the first study of inducibility in
tournaments.Comment: 11 pages, 5 figure
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