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

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.

• The OEIS sequence has a link to a table of the first 70 terms, and both Maple and Mathematica code to calculate more values. Aug 11, 2010 at 9:16
• I forget whom I learned it from, but it was someone here on MO and so I like to spread it here whenever possible, that the name is not Erdös but Erdős. I have edited accordingly. Feb 23, 2022 at 14:09
• Not exactly what you are asking, but certainly related: for finite groups $G$ of even order, the exact value of the constant is known; it is $2^{\nu(G)}-1$, where $\nu(G)$ is the number of even-order components in the canonical decomposition of $G$ into a direct sum of its cyclic subgroups: math.haifa.ac.il/seva/Papers/sfab.dvi
– Seva
Feb 23, 2022 at 21:09

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.