1 research outputs found
The Ramsey Number and Computational Bounds for
Using computer algorithms we establish that the Ramsey number
is equal to 37, which solves the smallest open case for Ramsey numbers of this
type. We also obtain new upper bounds for the cases of for , and show by construction a new lower bound .
The new upper bounds on are obtained by using the values and
lower bounds on for , where is the
minimum number of edges in any triangle-free graph on vertices without
in the complement. We complete the computation of the exact values of
for all with and for with ,
and establish many new lower bounds on for higher values of .
Using the maximum triangle-free graph generation method, we determine two
other previously unknown Ramsey numbers, namely and
. For graphs on 10 vertices, %besides ,
this leaves 6 other open besides , this leaves 6 open cases of the
form . The hardest among them appears to be , for which
we establish the bounds .Comment: 25 page