Original title: Výsledky v Manickam-Miklos-Singhiho domněnce
Translated title: Results in the Manickam-Miklos-Singhi Conjecture
Authors: Džavoronok, Adam ; Tyomkyn, Mykhaylo (advisor) ; Jelínek, Vít (referee)
Document type: Bachelor's theses
Year: 2026
Language: eng
Abstract: [eng] [cze]
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 domněnku, "e pro n alespo' 4k je odpově( )︄ n→1 k→1 [︄ . Pokrovskiy domněnku ově!il, kdy" n je alespo' 1046 k. P!ed ním Alon, Sudakov a Huang ově!ili, "e domněnka 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 zobecněními.
Keywords: additive combinatorics|hypergraph|MMS-property|extremal combinatorics; aditívní kombinatorika|hypergraf|MMS-vlastnost|extremální kombinatorika

Institution: Charles University Faculties (theses) (web)
Document availability information: Available in the Charles University Digital Repository.
Original record: http://hdl.handle.net/20.500.11956/211109

Permalink: http://www.nusl.cz/ntk/nusl-780439


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Universities and colleges > Public universities > Charles University > Charles University Faculties (theses)
Academic theses (ETDs) > Bachelor's theses