A Problem Concerning Nonincident Points and Blocks in Steiner Triple Systems

TL;DR

This paper investigates the maximum size of point and block sets in Steiner triple systems that are mutually nonincident, establishing an upper bound and conditions for equality.

Contribution

It introduces a new upper bound for the largest nonincident point-block sets in Steiner triple systems and demonstrates its attainability for infinitely many system orders.

Findings

Established an upper bound s  (2v+5 -   24v+25)/2

Proved the bound is tight for infinitely many v

Provides structural insights into nonincident configurations

Abstract

In this paper, we study the problem of finding the largest possible set of s points and s blocks in a Steiner triple system of order v, such that that none of the s points lie on any of the s blocks. We prove that s \leq (2v+5 - \sqrt{24v+25})/2. We also show that equality can be attained in this bound for infinitely many values of v.