13 research outputs found
New Computational Upper Bounds for Ramsey Numbers R(3,k)
Using computational techniques we derive six new upper bounds on the
classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <=
68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are
improvements by one over the previously best known bounds.
Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on
n vertices without independent sets of order k. The new upper bounds on R(3,k)
are obtained by completing the computation of the exact values of e(3,k,n) for
all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower
bounds on e(3,k,n) for most of the open cases for 10 <= k <= 15. The
enumeration of all graphs witnessing the values of e(3,k,n) is completed for
all cases with k <= 9. We prove that the known critical graph for R(3,9) on 35
vertices is unique up to isomorphism. For the case of R(3,10), first we
establish that R(3,10) = 43 if and only if e(3,10,42) = 189, or equivalently,
that if R(3,10) = 43 then every critical graph is regular of degree 9. Then,
using computations, we disprove the existence of the latter, and thus show that
R(3,10) <= 42.Comment: 28 pages (includes a lot of tables); added improved lower bound for
R(3,11); added some note
Ramsey numbers R(K3,G) for graphs of order 10
In this article we give the generalized triangle Ramsey numbers R(K3,G) of 12
005 158 of the 12 005 168 graphs of order 10. There are 10 graphs remaining for
which we could not determine the Ramsey number. Most likely these graphs need
approaches focusing on each individual graph in order to determine their
triangle Ramsey number. The results were obtained by combining new
computational and theoretical results. We also describe an optimized algorithm
for the generation of all maximal triangle-free graphs and triangle Ramsey
graphs. All Ramsey numbers up to 30 were computed by our implementation of this
algorithm. We also prove some theoretical results that are applied to determine
several triangle Ramsey numbers larger than 30. As not only the number of
graphs is increasing very fast, but also the difficulty to determine Ramsey
numbers, we consider it very likely that the table of all triangle Ramsey
numbers for graphs of order 10 is the last complete table that can possibly be
determined for a very long time.Comment: 24 pages, submitted for publication; added some comment
An exploration in Ramsey theory
We present several introductory results in the realm of Ramsey Theory, a subfield of Combinatorics and Graph Theory. The proofs in this thesis revolve around identifying substructure amidst chaos. After showing the existence of Ramsey numbers of two types, we exhibit how these two numbers are related. Shifting our focus to one of the Ramsey number types, we provide an argument that establishes the exact Ramsey number for h(k, 3) for k ≥ 3; this result is the highlight of this thesis. We conclude with facts that begin to establish lower bounds on these types of Ramsey numbers for graphs requiring more substructure