The largest (k, l)sum free subsets
Abstract
Let $\mathscr{M}_{(2,1)}(N)$ denotes the infimum of the size of the largest sumfree subset of any set of $N$ positive integers. An old conjecture in additive combinatorics asserts that there are a constant $c=c(2,1)$ and a function $\omega(N)\to\infty$ as $N\to\infty$, such that $cN+\omega(N)<\mathscr{M}_{(2,1)}(N)<(c+\varepsilon)N$ for any $\varepsilon>0$. The constant $c(2,1)$ is determined by Eberhard, Green, and Manners, while the existence of $\omega(N)$ is still widely open. In this paper, we study the analogue conjecture on $(k,\ell)$sum free sets and restricted $(k,\ell)$sum free sets. We determine the constant $c(k,\ell)$ for every $(k,\ell)$, and confirm the conjecture for infinitely many $(k,\ell)$.
 Publication:

arXiv eprints
 Pub Date:
 January 2020
 arXiv:
 arXiv:2001.05632
 Bibcode:
 2020arXiv200105632J
 Keywords:

 Mathematics  Combinatorics;
 Mathematics  Number Theory
 EPrint:
 25 pages, comments are welcome