The Erdős-Turán conjecture
Paul Erdős, one of the twentieth century’s most unique mathematicians, was known to travel from colleague to colleague, often arriving unannounced, and to immediately delve into some particularly intriguing research problem. See this article and this book for some additional background on this influential mathematician.
One of his more interesting conjectures is his “conjecture on arithmetic progressions,” sometimes referred to as the “Erdős-Turán conjecture.” It can be simply stated as follows: If $A$ is a set of positive integers such that $$\sum_{k \in A} \frac{1}{k} = \infty,$$ then $A$ contains arithmetic progressions of any length, or, in other words, $A$ contains subsets of the form $\{a, a+h, a+2h, a+3h, \cdots, a+(n-1)h\}$, for arbitrarily large $n$.
This conjecture was originally posed in a different form in 1936 — namely that any set of integers with positive natural density contains infinitely many three-term progressions. This was proven by Klaus Roth in 1952. In 1975, Szemeredi extended this result to arbitrarily long arithmetic progressions. Erdős posed the specific form of the conjecture given in the previous paragraph in 1976.
As of 2020, the conjecture remains unproven. Timothy Gowers of Cambridge, who received the Fields Medal in 1998, quipped, “I think many people regarded it as Erdős’ number-one problem.”
New breakthrough
A breakthrough has just been reported: Thomas Bloom of Cambridge and Olof Sisask of Stockholm University have proved the conjecture for triples, i.e., for arithmetic progressions of length three. In particular, their result is the following: If a set $A$ of positive integers contains no non-trivial three-term arithmetic progressions, then $|A|<< N/(\log N)^{1+c}$, for some absolute constant $c>0$. Or, stated another way, if $A$ is a set of positive integers such that $$\sum_{k \in A} \frac{1}{k} = \infty,$$ then $A$ contains three-term arithmetic progressions (infinitely many, in fact).
Several of the many mathematicians who have worked on variations of this problem over the years are impressed. Nets Katz of the California Institute of Technology, says, “This result was kind of a landmark goal for a lot of years.” “It’s a big deal.”
For full details, see the Bloom-Sisask paper (which has not yet been peer-reviewed) here. For an excellent overview of the problem and its history, including comments by several who have worked on the problem in the past, see this well-written Quanta Magazine article by Erica Klarreich.