Results in the Manickam-Miklos-Singhi Conjecture

Abstract

Suppose that we have a set S of n real numbers which have nonnegative sum. How few subsets of S of size k can have nonnegative sum? Manickam, Miklós and Singhi conjectured in 1987 that for n at least 4k the answer is )︄ n→1 k→1 [︄ . Pokrovskiy verified the conjecture for n at least 1046 k. Before that, Alon, Sudakov and Huang verified the conjecture for n at least 33k2 . In this thesis, we revisit these results and seek further improvements and generalisations.

P!edpokládejme, "e máme mno"inu S reáln#ch $ísel o velikosti n, jejich" sou$et je nezáporn#. Kolik podmno"in S o velikosti k m%"e mít nezáporn# sou$et? Manickam, Miklós a Singhi v roce 1987 vyslovili domn&nku, "e pro n alespo' 4k je odpov&( )︄ n→1 k→1 [︄ . Pokrovskiy domn&nku ov&!il, kdy" n je alespo' 1046 k. P!ed ním Alon, Sudakov a Huang ov&!ili, "e domn&nka platí, kdy" n je více ne" 33k2 . V této práci tyto v#sledky znovu prozkoumáme a pokusíme se p!ijít s dal)ími vylep)eními a zobecn&ními.