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 $c(S)>c_3$.

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