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Statement

Notation (p. 290): $\mathbb{N}=\{1,2,3,\ldots\}$.

Conjecture 11.5.6 (p. 291), credited to Erdős. If $A\subseteq\mathbb{N}$ satisfies $\sum_{a\in A}1/a=\infty$, then $A$ contains arbitrarily long arithmetic progressions.

The chapter places it among density versions of van der Waerden's theorem, after Szemerédi's theorem (Theorem 11.5.4, positive upper density suffices) and Gowers's quantitative form (Theorem 11.5.5), and follows it with Graham's two-dimensional analogue, Conjecture 11.5.7: a set $A\subseteq\mathbb{N}\times\mathbb{N}$ with $\sum_{(x,y)\in A}1/(x^2+y^2)=\infty$ contains the four vertices of an axis-aligned square.

Scope

This is a conjecture restated in a survey, not a result of the chapter.

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of J. E. Goodman, J. O'Rourke and C. D. Tóth (eds.), Handbook of Discrete and Computational Geometry, 3rd edition, CRC Press, Boca Raton, FL, 2017; the glossary and Theorem 11.5.4 on p. 290, Theorem 11.5.5 and Conjectures 11.5.6 and 11.5.7 on p. 291. Pages are those printed on the edition named on the source card.

Read depth. Claims checked: the statement was read on the printed page.

Bears on

• Problem 3: the conjecture is the problem's question stated as a conjecture. The chapter states it as a conjecture and contributes nothing toward it; the problem page records the later work.