Subset sums, completeness and colorings

Abstract

We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the three problems of Burr and Erdős on Ramsey complete sequences, for which Erdős later offered a combined total of $350; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and Erdős on the minimum number of colors needed to color the positive integers less than n so that n cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erdős and Graham on sets of integers avoiding a given subset sum; and, answering a question reiterated by several authors, a homogeneous strengthening of a seminal result of Szemerédi and Vu on long arithmetic progressions in subset sums.

Sufficient conditions for perfect mixed tilings

“…Given µ with χ + µ/2 ⌈χ⌉, we can assume that χ + µ/2 = (k − 1) + a/b for (coprime) natural numbers a b 8/µ. 6 We choose ̺ ′ , β > 0 such that…”

“…In this paper we adopt a result of Lev [25] to our setting. Questions of this type have recently been investigated by Conlon, Fox and Pham in [6].…”

The upper logarithmic density of monochromatic subset sums

“…In this section, we use a result from our recent paper [1] to prove the following key lemma on subset sums, which will be important in the proof of the lower bound for c 2 . We note that a weaker version of this lemma, from which the bound c 2 ≥ 1/2 easily follows, was previously claimed by Erdős [4,Theorem 3],…”

“…Theorem 3 (Theorem 6.1 of [1]). There exists an absolute constant C > 0 such that the following holds.…”

“…In fact, one can check that in this example each of Σ(A 1 ) and Σ(A 2 ) has upper logarithmic density 14/15. 1 Following this line of inquiry to its natural end, Erdős [3,4] asked for a determination of c 2 , the largest real number such that every two-coloring of the positive integers has a color class such that the upper logarithmic density of its set of subset sums is at least c 2 .…”