R(5,6) > 57

The two coloring of the edges of K57 which established the lower bound is saved here as a 0-1 adjacency matrix .

The construction requires three steps: first a set of five 19 by 19 circulant matrices is defined. Next, these five circulants are used in a 3 by 3 array to determine a coloring which is not good. Finally, 141 edges are recolored, giving the desired coloring which established the bound.

The five circulants are:

A  =  19(2,3,6,13,16,17)
B  =  19(3,4,6,9,10,13,15,16)
C  =  19(2,4,6,7,8,10,11,12,13,15,17)
D  =  19(6,7,11,13,15,16,17)
E  =  19(3,4,10,12,13,14,16,18)

These five 19 by 19 blocks are used as follows to form a 57 by 57 symmetric adjacency matrix :

A     C     D
C     A     E
D     E     B

The 141 edges whose colors need to be changed are listed below:

0-16       7-47      12-52      18-48      29-43      45-50
0-26       8-10      13-16      18-54      29-50 
0-33       8-35      13-20      19-22      30-44 
0-40       8-42      13-27      19-46      31-39 
1-27       8-51      13-53      20-38      32-40 
1-34       8-54      14-48      20-47      33-41 
1-41       9-20      15-17      20-51      33-45 
1-54       9-27      15-23      21-23      33-51 
2-20      10-21      15-39      21-39      34-42 
2-32      10-24      15-42      21-45      34-46 
2-35      10-27      15-49      21-52      34-52 
2-55      10-28      15-52      22-49      34-55 
3-21      10-40      16-27      23-44      35-40 
3-39      10-43      16-34      23-47      35-47 
3-56      10-46      16-56      23-56      35-49 
4-22      10-50      17-24      24-38      35-53 
4-38      10-53      17-28      25-28      36-50 
4-40      10-56      17-31      25-52      37-45 
5-19      11-22      17-34      26-53      37-51 
5-31      11-25      17-35      27-29      38-52 
5-41      11-29      17-38      27-39      39-44 
5-45      11-37      17-41      27-45      40-44 
5-54      11-41      17-44      27-48      40-48 
6-20      11-47      17-47      27-51      40-54 
6-32      11-51      17-53      27-54      41-45 
6-46      12-19      18-25      28-31      41-55 
7-21      12-26      18-32      28-55      42-53 
7-33      12-48      18-39      29-40      44-52 

Geoff Exoo