This has been bothering me for a while, and I can't seem to find any definitive answer. The following conjecture is well known in additive combinatorics:
Conjecture: If $A\subset \mathbb{N}$ and $$\sum_{a\in A} \frac{1}{a}$$ diverges, then $A$ contains arbitrarily long arithmetic progressions.
This is often called either "The Erdős-Turán conjecture" or "Erdős's conjecture on arithmetic progressions."
I want to make sure I quote things correctly in my writing, and my main question is, who should this conjecture be attributed to?
I am asking because I find a lot of conflicting pieces of information regarding how it should be referred to. For example, the Wikipedia page and the two Wolfram pages, page A and page B, are in disagreement. Wikipedia seems to be adamant that calling it the Erdős-Turán conjecture is incorrect, however I am skeptical, as it does not seem to be so clear. Also, both of the choices are used in many papers in the literature.
I went back and read the original 1936 paper of Erdős and Turán, "On some sequences of integers." (Math Sci Net MR1574918). In that paper, they conjecture that $r(N)=o(N)$, where $r(N)$ is a maximal subset of $\{1,\dots,N\}$ without 3 term arithmetic progressions. They also mention the (incorrect) conjectures of Szekeres on $k$ term progressions, but other than that, they don't seem to mention any conjectures regarding longer progressions at all.
As far as I can tell, the above conjecture first appears in the 1972 paper of Erdős, "Résultats et problèmes en théorie des nombres." (Math Sci Net MR0396376) This suggests that the conjecture should be named after Erdős, however, I believe that there must some reason why this conjecture has often been attributed to Turán as well. I can't imagine that it would be done for no reason, and it is entirely possible that the reason why credit is given to both Erdős and Turán was not written down, or at least, is not easy to find.
I am hoping someone can shed some light on this, and provide a clearer answer for what to call the above conjecture.
Thanks for your help,
Best Answer
One of the best places to track these things down is The mathematical coloring book, by Alexander Soifer, Springer 2009. Chapter 35 is on "Monochromatic arithmetic progressions", and section 35.4, "Paul Erdős’s Favorite Conjecture", is on the problem you ask about.
As far as I can tell, the question is sometimes called the Erdős-Turán conjecture for two reasons: First, it extends their older conjecture (now Szemerédi's theorem). Second, it was first popularized in connection to Turán, see below.
Soifer identifies a problem paper in the premier issue of Combinatorica as the first place in print were the conjecture appears with the attached value of $\$3000$: On the combinatorial problems which I would most like to see solved, Combinatorica, 1 (1981), 25–42, submitted Sep. 15, 1979.
Soifer adds that apparently Erdős first offered this amount in a 1976 talk, "To the memory of my lifelong friend and collaborator Paul Turán", in a conference the University of Manitoba, see Problems in number theory and Combinatorics, in Proceedings of the Sixth Manitoba Conference on Numerical Mathematics (Univ. Manitoba, Winnipeg, Man., 1976), Congress. Numer. XVIII, 35–58, Utilitas Math., Winnipeg, Man., 1977. (Erdős calls it "A very attractive conjecture of mine.")
In Problems and results on combinatorial number theory, II, J. Indian Math. Soc. (N.S.), 40(1–4) (1976), 285–298, he stated the conjecture with an attached prize of $\$2500$. In page 288 of this paper, we read (mildly edited for typos):
Here, as usual, $r_k(n)$ is the size of the largest subset of $\{1,2,\dots,n\}$ without $k$-term progressions.
By the way, the prize currently attached to the conjecture is actually $\$5000$. Soifer indicates that the update occurred in the paper Some of my favorite problems and results, in The Mathematics of Paul Erdős, vol. I, Graham, R. L., and Nesetril, J., (eds), 47–67, Springer, Berlin, 1997. (The paper appeared posthumously, and was written in 1996.)
Soifer then tries to identify when the conjecture was first stated.
The cited papers are:
For the finitude of $G_k$, see for example ArxiV:math/0703467.
By the way, the conjecture in the 1972 paper you identify seems to be restricted to $3$-term progressions. Even this version remains open (and carries an attached prize of $\$250$.)