Minimal number of edges in hypergraph guaranteeing perfect fractional matching and MMS conjecture, complete version

In this paper we prove Ahlswede- Khachatrian conjecture~\cite{1} up to finite number of cases, which can be checked using modern computers. From this conjecture follows conjecture from~\cite{2} and Manickam-Mikl\'{o}s-Singhi conjecture. This paper is combined from papers~\cite{09},~\cite{08}.