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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 ErdH{o}s on Ramsey complete sequences, for which ErdH{o}s 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 ErdH{o}s 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 ErdH{o}s 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 Szemeredi and Vu on long arithmetic progressions in subset sums.
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We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least $(2+sqrt{3})/4$ and this is best possible. This answers a forty-year-old question of ErdH{o}s.
Let $vec{w} = (w_1,dots, w_n) in mathbb{R}^{n}$. We show that for any $n^{-2}leepsilonle 1$, if [#{vec{xi} in {0,1}^{n}: langle vec{xi}, vec{w} rangle = tau} ge 2^{-epsilon n}cdot 2^{n}] for some $tau in mathbb{R}$, then [#{langle vec{xi}, vec{w} rangle : vec{xi} in {0,1}^{n}} le 2^{O(sqrt{epsilon}n)}.] This exponentially improves the $epsilon$ dependence in a recent result of Nederlof, Pawlewicz, Swennenhuis, and Wk{e}grzycki and leads to a similar improvement in the parameterized (by the number of bins) runtime of bin packing.
Some class of sums which naturally include the sums of powers of integers is considered. A number of conjectures concerning a representation of these sums is made.
Let $G$ be an additive abelian group and $Ssubset G$ a subset. Let $Sigma(S)$ denote the set of group elements which can be expressed as a sum of a nonempty subset of $S$. We say $S$ is zero-sum free if $0 otin Sigma(S)$. It was conjectured by R.B.~Eggleton and P.~Erd{o}s in 1972 and proved by W.~Gao et. al. in 2008 that $|Sigma(S)|geq 19$ provided that $S$ is a zero-sum free subset of an abelian group $G$ with $|S|=6$. In this paper, we determined the structure of zero-sum free set $S$ where $|S|=6$ and $|Sigma(S)|=19$.
For a subset A of a finite abelian group G we define Sigma(A)={sum_{ain B}a:Bsubset A}. In the case that Sigma(A) has trivial stabiliser, one may deduce that the size of Sigma(A) is at least quadratic in |A|; the bound |Sigma(A)|>= |A|^{2}/64 has recently been obtained by De Vos, Goddyn, Mohar and Samal. We improve this bound to the asymptotically best possible result |Sigma(A)|>= (1/4-o(1))|A|^{2}. We also study a related problem in which A is any subset of Z_{n} with all elements of A coprime to n; it has recently been shown, by Vu, that if such a set A has the property Sigma(A) is not Z_{n} then |A|=O(sqrt{n}). This bound was improved to |A|<= 8sqrt{n} by De Vos, Goddyn, Mohar and Samal, we further improve the bound to the asymptotically best possible result |A|<= (2+o(1))sqrt{n}.
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