TITLE: A reformulation of Erdős conjecture on arithmetic progressions
QUESTION [12 upvotes]: Erdős conjecture on arithmetic progressions states that if $S$ is a set of positive integers such that $c(S):=\sum_{n \in S} \frac{1}{n} = \infty$ (large set), then $ \forall \ell \ge 3$ the set $S$ contains an arithmetic progression of length $\ell$. In other words, this conjecture states that for all set $S$ of positive integers, if $\exists \ell \ge 3$ such that $S$ contains no arithmetic progression of length $\ell$, then $c(S) < \infty$.  
Let $\ell \ge 3$ and $S_{\ell}$ be the lexicographically earliest (strictly) increasing sequence of positive integers that contains no arithmetic progression of length $\ell$. We identify $S_{\ell}$ with its set of terms. According to Erdős conjecture, we should have $c_{\ell}:=c(S_{\ell}) < \infty$. But intuitively, $S_{\ell}$ is the worst set with no arithmetic progression of length $\ell$ to deal with Erdős conjecture. This leads to:  
Main Question: Is Erdős conjecture equivalent to $c_{\ell} < \infty$ for all $\ell \ge 3$?
Let $\ell \ge 3$ and $S$ be a set of positive integers with no arithmetic progression of length $\ell$.    
Stronger Question 1: Is it true that $c(S) \le c_{\ell}$?
Answer: No, see the comment of Andrés E. Caicedo citing this paper of J. Wróblewski (see below).    
Stronger Question 2: Is it true that $c(S) < c_{\ell}$ or $c(S) \sim c_{\ell}$?   

Note that OEIS includes the sequences $S_{\ell}$ for all $\ell \in \{3,4,\dots,10 \}$:  

$S_3$: $1, 2, 4, 5, 10, 11, 13, 14, 28, 29, 31, 32, 37, 38, 40, \dots$ (A003278)
$S_4$: $ 1, 2, 3, 5, 6, 8, 9, 10, 15, 16, 17, 19, 26, 27, 29, 30, \dots$ (A005837)
$\vdots$ 
$S_{10}$: $1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 12, 13, 14, 15, 16, 17, 18, \dots$ (A020664)

Ralf Stephan put the following formula in A003278:
If $S_3(n)=b(n+1)$ then $b(0)=1$, $b(2n)=3b(n)-2$ and $b(2n+1)=3b(n)-1$.
It follows that $S_3(n)$ grows like $n^{\alpha}$ with $\alpha:=\frac{\log(3)}{\log(2)}$, so that $c_3<\infty$.  
Moreover, $c_3 = \sum_{n=1}^{N}S_3(n)^{-1}+ O(N^{1-\alpha})$. We computed (after the comment of Bullet51) $∑_{n=1}^{10^8}S_3(n) = 3.007886\dots$, whereas $10^{8(1-\alpha)} = 0.0000209\dots$.
Now, Wróblewski's paper states that $ 3.00793<c_3<3.00794.$ 
Bonus Question: What is the exact value of $c_3$?
We also computed (using OEIS b-files) $\sum_{n=1}^{10^5}S_4(n)^{-1} = 4.1911..$, $\sum_{n=1}^{10^5}S_{10}(n)^{-1} = 7.7326..$, but we don't know how good they are as approximations of $c_4$ and $c_{10}$ respectively.
Bonus Question ${\ell}$: What is a good approximation and/or the exact value of $c_{\ell}$?  
The above recursive formula for $S_3(n)$ matches with its graph of "Cantor flavor". Now for $\ell >3$, the graph seems to lose this flavor (see here), so that we can no more expect such a recursive formula.
Question of Sylvain Julien: Is $c_{\ell}$ an increasing function of $\ell$?
More strongly: Is it true that $ℓ≤ℓ′$ implies $S_ℓ(n)≥S_{ℓ′}(n)$ for all $n≥1$?  

Wróblewski's paper provides a set $S$ of positive integers with no arithmetic progression of length $3$ and with $c(S)>3.00849$. That's why the answer to Stronger Question 1 is no, but the Stronger Question 2 and Main Question are still open (to me).   
Let $\mathcal{A}_{\ell}$ be the set of sets of positive integers with no arithmetic progression of length $\ell$, and let $a_{\ell}:=\sup_{S \in \mathcal{A}_{\ell}}(c(S))$. Now my belief is that $a_{\ell} < \infty$, that $a_{\ell} \sim c_{\ell}$, but also that there is no $S \in \mathcal{A}_{\ell}$ with $c(S) = a_{\ell}$. I believe that a proof of $a_3<\infty$ should be hard but reachable, on the other side, I believe that the exact value of $a_3$ or even a good approximation (with say 10 digits) is not reachable with the current knowledge/technics. What could be reachable is to break records, starting by the above old record (July 1984) of Wróblewski.     
Is Wróblewski's record broken today?  If yes:
What is the largest $c(S)$ known today, with $S \in \mathcal{A}_3$? (same question for every fixed $\ell > 3$)
Thomas' answer suggests a set $S$ of Moser (1953), but we don't even know if $c(S)>c_3$.

REPLY [2 votes]: I asked Prof. Jaroslaw Wróblewski by email, below is his answer (reproduced with his authorization):  

I do not know of any new results regarding searching a set $B$ in $\mathcal{A}_3$ [the set of sets of positive integers with no arithmetic progression of length $3$]
  with large $c(B)$. However since I wasn't interested in the subject for
  years, I may not be a good source of information. I am sure however,
  that such a set can be constructed easily with help of nowadays
  computers - my 35 year old construction was made virtually by hand and
  a limited access to a computer was used only to get a good estimate of
  the sum of reciprocals.
I have been collecting results regarding large (i.e. having many
  elements) non-avereging finite sets and the best results known to me
  are here: http://www.math.uni.wroc.pl/~jwr/non-ave.htm
As for $a_3$ [$:=\sup_{S \in \mathcal{A}_{3}}(c(S))$], I am afraid it is impossible to have an
  educated guess on that. Note that  Szekeres's sequence seems to be the
  best possible before you see a construction which can improve it. So I
  may give only my speculations which have value of tea leaves reading:

I feel that most likely  $a_3$ is finite, but only slightly larger than $3$.
Less likely, but reasonable is that  $a_3 = \infty$.
I feel that it is rather unlikely that $a_3$ is finite, but significantly larger than $3$. The reason for that is the
  following: Szekeres's sequence is the best one for a quite long run.
  There are sets that are better, but they are better far away (on large
  numbers). If they give finite sum of reciprocals, then this is sum of
  reciprocals of large numbers, so it is likely to be very small
  (assuming it is finite) - most accidental sets of large numbers have
  small or infinite sum of reciprocals - a set of large numbers with
  large but finite sum of reciprocals can be easily constructed (must
  have a very specific density of terms), but is rather unlikely to
  appear accidentally as a set arising in a problem not direcly related
  to such a sum.