### The 16 element relation algebra with the longest proof of nonrepresentability

(still under construction...:-)

a is red, b is blue, c is green

Consistent triples (must appear on each matching edge in a representation)

Inconsistent triples (must NOT appear in any representation)
Game theoretic proof that this relation algebra is not representable:

N1 = c12 b13a23 a14b24c34 A:a15a52

N2 = c12 a13b23 a14b24c34 A:a15a52
N3 = c12 b13c23 b14a24b34 a15b25c35b45 A:a16a62 [ 2, 2, 3, 4 ], N2[ 2, 3, 4, 6 ]

N4 = c12 c13a23 a14c24b34 a15b25c35c45 A:a16a62
N5 = c12 c13a23 b14c24a34 b15a25c35c45 A:a16a62
N6 = c12 c13b23 b14c24a34 b15c25c35b45 A:a16a62 [ 2, 2, 3, 3, 3 ], N3[ 2, 3, 4, 5, 6 ]
[ 2, 2, 3, 3, 4 ], N3[ 1, 3, 4, 5, 6 ]
[ 2, 2, 3, 4 ], N1[ 1, 3, 4, 6 ]
[ 2, 2, 4, 3 ], N1[ 2, 3, 4, 6 ]
[ 2, 2, 4, 4 ], N4[ 1, 2, 3, 4, 6 ]

N7 = b12 c13c23 c14c24b34 a15b25b35b45 A:a16c62 [ 2, 4, 2, 3, 4 ], N2[ 1, 4, 5, 6 ]
[ 2, 4, 3, 2, 4 ], N2[ 1, 3, 5, 6 ]
[ 2, 4, 3, 3 ], N6[ 1, 2, 3, 4, 6 ]

N8 = b12 b13c23 b14b24a34 a15b25c35b45 A:a16c62 [ 2, 4, 3, 3, 2 ], N3[ 2, 3, 4, 5, 6 ]
[ 2, 4, 3, 3, 4 ], N7[ 2, 3, 4, 5, 6 ]
[ 2, 4, 3, 4 ], N3[ 1, 2, 3, 4, 6 ]

N9 = b12 c13a23 b14c24c34 b15a25c35b45 A:a16c62 [ 2, 4, 2, 3, 3 ], N8[ 1, 2, 4, 5, 6 ]
[ 2, 4, 2, 3, 4 ], N5[ 2, 3, 4, 5, 6 ]
[ 2, 4, 3 ], N1[ 1, 2, 3, 6 ]
[ 2, 4, 4, 3, 3 ], N8[ 1, 2, 4, 5, 6 ]

N10 = c12 b13c23 b14c24b34 b15a25a35b45 A:a16b62 [ 2, 3, 3, 3, 3 ], N8[ 1, 2, 4, 5, 6 ]
[ 2, 3, 3, 3, 4 ], N1[ 1, 2, 5, 6 ]
[ 2, 3, 3, 4, 3 ], N3[ 1, 2, 4, 5, 6 ]
[ 2, 3, 3, 4, 4 ], N1[ 1, 2, 5, 6 ]
[ 2, 3, 4, 3, 3 ], N8[ 1, 2, 4, 5, 6 ]
[ 2, 3, 4, 3, 4 ], N1[ 1, 2, 5, 6 ]
[ 2, 3, 4, 4 ], N6[ 1, 2, 3, 4, 6 ]

N11 = b12 b13c23 b14c24b34 b15a25a35b45 A:a16c62 [ 2, 4, 3, 3, 3 ], N8[ 1, 2, 4, 5, 6 ]
[ 2, 4, 3, 3, 4 ], N9[ 2, 3, 4, 5, 6 ]

N12 = b12 c13c23 b14a24a34 a15b25a35c45 A:b16b62 [ 3, 3, 2, 4 ], N2[ 1, 3, 4, 6 ]
[ 3, 3, 3, 3 ], N11[ 1, 2, 3, 4, 6 ]
[ 3, 3, 3, 4 ], N2[ 2, 3, 4, 6 ]
[ 3, 3, 4, 3 ], N10[ 1, 2, 3, 4, 6 ]
[ 3, 3, 4, 4 ], N9[ 1, 2, 3, 4, 6 ]

G(0) = [ [ 4 ], [ 3, 2 ], [ 2, 3, 4 ], [ 2, 2, 4, 4 ], [ 3, 2, 4, 4, 4 ] ]
c12 b13a23 a14b24b34 a15a25c35c45 A:b16a62[ 1, 2, 3, 4, 5 ]12
c12 b13a23 a14b24c34 a15a25c35c45 A:b16a62[ 1, 2, 3, 4 ]1