# On the constants in the Cameron-Erdös conjecture on sum-free subsets.

The Cameron-Erdös conjecture was proved independently by Ben Green and Alexander Sapozhenko.

Let s(n) be the number of sum-free subsets of the set of integers {1,2,...,n}. They showed that ${ s(n) / 2^{n/2} } \to C_O \hbox{ or } C_E,$ for constants $C_O$ and $C_E$, as $n \to \infty$ through odd or even values respectively.

I would like to know what are the best known bounds for the constants $C_O$ and $C_E$?

My motivation is that I considered the conjecture in the mid-1990s and tried to determine some good lower bounds for the constants on the condition that the limits existed, of course. I have a vague recollection that Cameron and Erdös had some lower bounds in the region of 5 or 6, but I no longer have their relevant papers handy to verify this.

Looking at the sequence A007865 in the OEIS, it would seem that $C_E$ is in the region of 13.4 and that $C_O$ is in the region of 14.4. If one calculates at $s(n)/2^{n/2}$ for even n, it rises steadily from n=0 to n=36 then interestingly appears to oscillate about its limit. The sequence for odd n, from n=39 onwards, possibly does the same. It would be interesting to have some more terms.

Anyway, any information that you have on the actual values of these constants would be greatly appreciated.

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The OEIS sequence has a link to a table of the first 70 terms, and both Maple and Mathematica code to calculate more values. – Hugo van der Sanden Aug 11 '10 at 9:16
Many thanks Hugo. – Derek Jennings Aug 11 '10 at 12:31

The review of the Sapozhenko paper, The Cameron-Erdos conjecture, Dokl. Akad. Nauk 393 (2003) 749-752, MR 2006a:11027, says $s(n)$ is asymptotic to $(c_0+1)2^{\lceil n/2\rceil}$ when $n$ is even and $(c_1+1)2^{\lceil n/2\rceil}$ when $n$ is odd, with $4.036\le c_0\le4.079$ and $3.086\le c_1\le3.095$.
EDIT: The review of a more recent paper, K G Omel'yanov, Estimates for Cameron-Erdos constants, Diskret. Mat. 18 (2006) 55-70, translation in Discrete Math. Appl. 16 (2006) 205-220, MR 2007m:11038, seems to contradict these numbers, giving $5.0709\le c_0\le5.0995$ and $3.8103\le c_1\le3.8336$. I haven't looked at the primary sources, so am unable to say whether the problem lies with me, a reviewer, or an author.