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Statement

Conjecture 11.5.6 (p. 10, attributed to Erdős, quoted). "If $A\subseteq\mathbb N$ satisfies $\sum_{a\in A}1/a=\infty$ then $A$ contains arbitrarily long arithmetic progressions."

Here $\mathbb N=\{1,2,3,\ldots\}$ (p. 10). The chapter places it among conjectured density versions of van der Waerden's theorem that use a measure of size other than positive upper density.

Context on the same page (p. 10). Szemerédi's theorem (Theorem 11.5.4): a set $A\subseteq\mathbb N$ of positive upper density contains arbitrarily long arithmetic progressions. Gowers's bound (Theorem 11.5.5, [Gow01]): for every $k>0$, every subset of $\{1,2,\ldots,N\}$ of size at least $N(\log\log N)^{-c(k)}$ contains a $k$-term arithmetic progression, where $c(k)=2^{-2^{k+9}}$. Graham's Conjecture 11.5.7: if $A\subseteq\mathbb N\times\mathbb N$ and $\sum_{(x,y)\in A}1/(x^2+y^2)=\infty$, then $A$ contains the four vertices of an axes-parallel square; the chapter expects, more generally, a homothetic image of $\{1,\ldots,m\}\times\{1,\ldots,m\}$ for every $m$.

Source. R. L. Graham, Euclidean Ramsey theory, Chapter 11 of the Handbook of Discrete and Computational Geometry, 2nd edition, CRC Press (2004), read in the preprint of the chapter identified on the source card, whose own page number is cited: all of the above on p. 10.

Read depth. Claims checked: the statements were read clause by clause on the page image of the preprint. The conjecture is stated without proof, and the cited theorems were not checked here. Nothing here is independently reviewed.

Proof pointer

None; the chapter states the conjecture as open.

Dependencies

None.

Bears on

• Problem 3: Conjecture 11.5.6 is the problem's question, stated as a conjecture of Erdős; the chapter proves nothing toward it.