On the constants in the Cameron–Erdős conjecture on sum-free subsets

Question

The Cameron–Erdős conjecture was proved independently by Ben Green (The Cameron-Erdos Conjecture) and Alexander Sapozhenko (The Cameron-Erdős conjecture).

Let $s(n)$ be the number of sum-free subsets of the set of integers $\{1,2,\dotsc,n\}$. They showed that ${ s(n) / 2^{n/2} } \to C_O \text{ or } C_E$, 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 Erdős 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 $s(n)/2^{n/2}$ 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.

Answer 1

The review of the Sapozhenko paper, The Cameron–Erdős 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–Erdős 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.