# Ramsey Numbers

• An Improved Lower Bound for R(3,11) was found in November 2023. An adjacency list for this coloring can be found here. The lower bound is now 47.
• 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 lower bound for R(5,6) was improved from 58 to 59 (October 2023). The coloring is available here. A short note on the construction is available here.
• 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.
• The lower lound for the 3-graph 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.
• The lower lound for the 3-graph 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.

## Two Color Ramsey Colorings from the Dynamic Survey.

The following table gives some of my constructions for bounds that appear in Table I of the Dynamic Survey, plus a few others. Note that the last two entries in the table are for Ramsey Numbers of 3-uniform hypergraphs.
 R(5,5) > 42 R(3,10) > 39 R(3,11) > 46 R(3,16) > 81 R(3,18) > 98 R(3,20) > 110 R(4,6) > 35 R(4,7) > 48 R(4,9) > 48 R(5,6) > 58 R(4,5;3) > 32 R(5,5;4) > 33

## Multicolor Ramsey Numbers

Here are some colorings obtained using the "growth method", a simple technique described in a paper 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 j-i.

 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 R(3,4,5) > 88 Matrix

## 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.