Generalized Schreier sets, linear recurrence relation, Tur\'{a}n graphs

TL;DR

This paper establishes a linear recurrence relation for generalized Schreier sets, extending Fibonacci-type recurrences, and explores their connection to Turán graphs, revealing new structural insights.

Contribution

It introduces a generalized recurrence relation for Schreier sets and links these sets to Turán graphs, broadening understanding of their combinatorial properties.

Findings

Derived a linear recurrence for generalized Schreier sets

Connected Schreier sets to Turán graphs

Extended Fibonacci recurrence to higher orders

Abstract

We prove a linear recurrence relation for a large family of generalized Schreier sets, which generalizes the Fibonacci recurrence proved by Bird and higher order Fibonacci recurrence proved by the second author et al. Furthermore, we show a relationship between Schreier-type sets and Tur\'{a}n graphs.