185 research outputs found
Chromatic number of graphs and edge Folkman numbers
In the paper we give a lower bound for the number of vertices of a given
graph using its chromatic number. We find the graphs for which this bound is
exact. The results are applied in the theory of Foklman numbers.Comment: 9 pages, 1 figur
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
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
On Some Generalized Vertex Folkman Numbers
For a graph and integers , the expression means that for any -coloring of the vertices of there
exists a monochromatic -clique in for some color . The vertex Folkman numbers are defined as
is -free and , where is a graph. Such vertex Folkman numbers have
been extensively studied for with . If
for all , then we use notation .
Let be the complete graph missing one edge, i.e. . In
this work we focus on vertex Folkman numbers with , in particular for
and . A result by Ne\v{s}et\v{r}il and R\"{o}dl from 1976
implies that is well defined for any . We present a new
and more direct proof of this fact. The simplest but already intriguing case is
that of , for which we establish the upper bound of 135. We
obtain the exact values and bounds for a few other small cases of
when for all , including
, , and . Note
that is the smallest number of vertices in any -free graph
with chromatic number . Most of the results were obtained with the help of
computations, but some of the upper bound graphs we found are interesting by
themselves
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