Below we describe a two coloring of the edges of K34 containing no monochromatic K4 in the first color, and no monochromatic K6 in the second. Note: in describing the coloring we refer to colors 0 and 1, and vertices 0 to 33.
The construction is made in three stages: first a set of four 2 by 2 matrices is defined. Next, these four matrices are used to construct in a 17 by 17 array (determining a coloring of K34 which contains several monochromatic K4's). Finally, 11 edges are recolored.
The four 2 by 2 matrices are:
A B C D
0 0 1 0 0 1 1 1
0 0 0 1 1 0 1 1
These four 2 by 2 blocks are used as follows to form a 34 by 34 symmetric adjacency matrix (we show only the upper triangle):
A D B B A C C D B D C A B D C D C
_ C C C C B B D B D B A B A C D C
_ _ C D B D D A C D D C D B C A A
_ _ _ A B D D B B B A C C C D B D
_ _ _ _ C C B D C A B D D D C B B
_ _ _ _ _ A A B D B D C C B D B D
_ _ _ _ _ _ C C D B B B B C D B D
_ _ _ _ _ _ _ C D B D C B D D C A
_ _ _ _ _ _ _ _ A D D D C C A B C
_ _ _ _ _ _ _ _ _ C C B B A B B B
_ _ _ _ _ _ _ _ _ _ C C C C C C D
_ _ _ _ _ _ _ _ _ _ _ C B D B B D
_ _ _ _ _ _ _ _ _ _ _ _ A D C D A
_ _ _ _ _ _ _ _ _ _ _ _ _ A C B C
_ _ _ _ _ _ _ _ _ _ _ _ _ _ C C B
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ A D
_ _ _ _ _ _ _ _ _ _ _ _ _ _ _ _ C
The matrix F gives a coloring of K34 with exactly 2 monochromatic K4's in color zero and no K6's in color one. To complete the construction the colors of the following 11 edges are changed. Note again that the vertices are numbered from zero, rather than from one.
1-26 6-14 7-15 10-14 10-15
12-14 12-15 18-23 19-22 24-25
30-31
Geoff Exoo