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Statement

Properties C and DW are defined on the definitions page.

Theorem 3 (p. 5, quoted). "If $A$ is a set of positive integers with infinite reciprocal sum, then $A$ has property $C$ (and therefore also property $DW$)."

Source. Brown, T. C., Erdős, P. and Freedman, A. R., Quasi-progressions and descending waves, J. Combin. Theory Ser. A 53 (1990), no. 1, 81--95, doi:10.1016/0097-3165(90)90021-N, read in the authors' copy identified on the source card, whose pages are numbered 1 to 13: the statement on p. 5, the proof on pp. 5--6.

Read depth. Claims checked: the statement was read on the print's page. The proof was read but not checked step by step. Nothing here is independently reviewed.

Proof pointer

Section 3, pp. 5--6. A density bound for cubes, which the introduction calls Szemerédi's method for obtaining cubes, says that with $\alpha=2+\sqrt3$ any subset of $\{1,\ldots,n\}$ with at least $\alpha n^{1-1/2^k}$ elements contains a $k$-cube. So a set with no $k$-cube has counting function below $\alpha n^{1-1/2^k}$, its $n$th element grows at least like $cn^{1+\epsilon}$ for positive constants $c,\epsilon$, and its reciprocal sum converges. The parenthetical clause follows from $C\Rightarrow DW$ in Theorem 1.

Dependencies

The cube density bound, cited from R. L. Graham, Rudiments of Ramsey theory, Amer. Math. Soc., 1981, p. 19 (p. 5); the implication $C\Rightarrow DW$ of Theorem 1. A second proof that infinite reciprocal sum gives property DW, independent of this theorem, is in the remarks after Corollary 2.

Bears on

• Problem 3: the theorem proves the problem's assertion with arbitrarily long arithmetic progressions weakened to arbitrarily large cubes, a strictly weaker property by Theorem 1; it does not give progressions.