Ramsey Numbers
Note: An Improved Lower Bound for R(3,16) was found in March 2013. An
adjacency matrix for this coloring can be found here.
The coloring is a Cayley coloring, using the group identified as
SmallGroup(81,5) in GAP.
This improves the lower bound from 79 to 82.
Note: The lower lound for the 3graph Ramsey
number R(5,5;3) has been improved from 65 to 82 (February 2013).
One of the (many) colorings can be found here.
The coloring can be verified with this program.
This program was written more for transparency that for efficiency.
Note: The lower lound for the 3graph Ramsey
number R(4,6;3) has been improved from 38 to 58 (November 2012).
One of the (many) colorings can be found here.
The coloring can be verified with this program.
This program was written more for transparency that for efficiency.
Note: An Improved Lower Bound for R(3,11) was found in November 2012. An
adjacency list for this coloring can be found here.
The lower bound is now 47.
Note: An Improved Lower Bound for R(4,8) was found in December 2012. An
adjacency list for this coloring can be found here.
The lower bound is now 58.
Note: An Improved Lower Bound for R(4,6) was found in March 2012. A brief
description of one coloring can be found here.
A list of 37 new colorings can be found here.
The first table gives some new (July 2005) lower bounds for Ramsey
numbers. These improve the values given in the most recent edition of the
Dynamic Survey. The colorings will be available soon.
New Lower Bounds for Classical Ramsey Numbers from Cayley Colorings
R(3,18)  >  98 
R(3,20)  >  110 
R(4,9)  >  72 
R(4,16)  >  162 
R(5,9)  >  124 
R(5,10)  >  142 
R(5,11)  >  158 
R(5,12)  >  184 
R(5,13)  >  208 
R(5,14)  >  234 
R(5,15)  >  264 
R(6,7)  >  112 
R(3,3,7)  >  80 
R(3,3,8)  >  100 
R(3,3,9)  >  116 
R(3,4,5)  >  80 
R(3,4,6)  >  106 
R(3,4,7)  >  142 
R(3,5,5)  >  128 
R(5,5,5)  >  416 

This rest of this page gathers links to older constructions for
various Ramsey numbers.
The constructions
given here are mine; for
a more complete list
see the
Dynamic Survey.
Older Lower Bounds for Multicolor Ramsey Numbers
Here are a new links
to colorings obtained using the
"growth method", a simple technique I described
in
Volume 1 of the Electronic Journal of
Combinatorics.
Note that the colorings are not circle colorings.
They might be called "linear" colorings, in that for
i < j,
the color of the edge joining vertex i to vertex j depends
only on ji.
R(4,4,4,4)  >  577 

Matrix (gzipped) 

Chords 
R(5,5,5)  >  414 

Matrix (gzipped) 

Chords 
R(3,3,3,4)  >  92 

Matrix 

Chords 
R(3,3,4,4)  >  170 

Matrix 

Chords 
R(3,3,7)  >  78 

Matrix 

Chords 

Ramsey Colorings from Noncyclic Groups
The next group of links below are to
files containing adjacency matrices for
Ramsey colorings of complete graphs.
Most of these
establish best known lower bounds
for classical Ramsey numbers.
They are all new and were made
using nonabelian groups of order pq (for
primes p and q) using
two different techniques.
The coloring for R(4,8) is a Cayley
coloring, i.e., both color graphs
are Cayley graphs. The underlying group is
the nonabelian group of order 55.
The latter three colorings were obtained
using a technique described
here.
More recently, colorings that give new lower bounds
for R(4,9) and R(3,20) were found using a broader
class of nonabelian groups.
Old Ramsey Colorings from the Dynamic Survey
Here are some links to some of the constructions for
classical Ramsey numbers that are referenced in
the Dynamic Survey.
R(5,5) 
>  42 

R(4,6) 
>  35 

R(3,10) 
>  39 

R(3,11) 
>  46 
R(3,12) 
>  51 

R(4,7) 
>  48 

R(5,6) 
>  57 

R(5,9) 
>  115 
R(5,10) 
>  140 

R(5,11) 
>  152 

R(5,12) 
>  180 

R(5,13) 
>  192 
R(5,14) 
>  220 

R(5,15) 
>  236 

R(6,7) 
>  108 

R(6,8) 
>  121 
R(6,9) 
>  152 

R(6,10) 
>  184 

R(3,4,5) 
>  79 

R(4,5;3) 
>  32 
R(5,5;4) 
>  33 


Complete Graphs Missing One Edge
Finally, here is
paper which describes some of the entries from Table III of the
Dynamic Survey. The graphs can be found here.
Geoff Exoo