352 research outputs found
A Folkman Linear Family
For graphs and , let signify that any red/blue edge
coloring of contains a monochromatic . Define Folkman number to
be the smallest order of a graph such that and . It is shown that for graphs of order with
, where , and are
positive constants.Comment: 11 page
Combinatorial theorems relative to a random set
We describe recent advances in the study of random analogues of combinatorial
theorems.Comment: 26 pages. Submitted to Proceedings of the ICM 201
Improved bounds on the multicolor Ramsey numbers of paths and even cycles
We study the multicolor Ramsey numbers for paths and even cycles,
and , which are the smallest integers such that every coloring of
the complete graph has a monochromatic copy of or
respectively. For a long time, has only been known to lie between
and . A recent breakthrough by S\'ark\"ozy and later
improvement by Davies, Jenssen and Roberts give an upper bound of . We improve the upper bound to . Our approach uses structural insights in connected graphs without a
large matching. These insights may be of independent interest
Ramsey-nice families of graphs
For a finite family of fixed graphs let be
the smallest integer for which every -coloring of the edges of the
complete graph yields a monochromatic copy of some . We
say that is -nice if for every graph with
and for every -coloring of there exists a
monochromatic copy of some . It is easy to see that if
contains no forest, then it is not -nice for any . It seems
plausible to conjecture that a (weak) converse holds, namely, for any finite
family of graphs that contains at least one forest, and for all
(or at least for infinitely many values of ),
is -nice. We prove several (modest) results in support of this
conjecture, showing, in particular, that it holds for each of the three
families consisting of two connected graphs with 3 edges each and observing
that it holds for any family containing a forest with at most 2
edges. We also study some related problems and disprove a conjecture by
Aharoni, Charbit and Howard regarding the size of matchings in regular
3-partite 3-uniform hypergraphs.Comment: 20 pages, 2 figure
- β¦