The (7,4)-conjecture in finite groups

TL;DR

This paper proves the (7,4)-conjecture for finite groups, establishing a threshold condition under which large subsets of triples in a finite group guarantee a specific combinatorial configuration.

Contribution

It provides the first proof of the (7,4)-conjecture for finite groups, advancing understanding of combinatorial structures in algebraic systems.

Findings

Confirmed the (7,4)-conjecture for finite groups.

Established a threshold condition for the existence of specific triples.

Extended combinatorial conjectures to algebraic structures.

Abstract

The first open case of the Brown, Erd\H{o}s, S\'os conjecture is equivalent to the following; For every $c>0$ there is a threshold $n_0$ so that if a quasigroup has order $n\geq n_0$ then for every subset of triples of the form $(a,b,ab),$ denoted by $S,$ if $|S|\geq cn^2$ then there is a seven-element subset of the quasigroup which spans at least four triples of the selected subset $S.$ In this paper we prove the conjecture for finite groups.