A construction of Steiner Triple Systems of type $v\longrightarrow 2v+7$

TL;DR

This paper presents a difference method construction for Steiner Triple Systems of type v→2v+7, enabling the creation of larger systems with specific properties from smaller ones, for all n≥5.

Contribution

It introduces a new recursive construction method for Steiner Triple Systems of type v→2v+7 based on difference techniques, expanding the class of known systems.

Findings

Constructs STS of order 2^{n+1}-7 from STS of order 2^n-7

Produces systems with maximal independent sets of maximal size

Ensures systems are (n-1)-bicolorable for n≥5

Abstract

A Steiner Triple System ($STS$) of order $v$ is a hypergraph uniform of rank 3, with $v$ vertices and such that every 2-subset of vertices has degree 1. In this paper we give a construction, by difference method, of type $v\longrightarrow 2v+7$ with $v=2^n-7$, which means that, given an $STS$ of order $v=2^n -7$, it is always possible to construct an $STS$ of order $2^{n+1}-7$. Through this construction it is possible to get for any $n\ge 5$ an $STS(2^n-7)$ with a maximal independent set of maximal cardinality and which is $(n-1)$-bicolorable.