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

TL;DR

This paper proves a significant combinatorial conjecture for most cases using computational methods, with implications for related conjectures in hypergraph theory and combinatorics.

Contribution

It verifies the Ahlswede-Khachatrian conjecture for all but finitely many cases, advancing understanding of hypergraph matchings and related conjectures.

Findings

Confirmed the Ahlswede-Khachatrian conjecture in most cases

Connected the conjecture to the Manickam-Miklós-Singhi conjecture

Utilized computational checks for finite cases

Abstract

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}.