Two extremal problems in the light of Lex graphs

TL;DR

This paper demonstrates that two key extremal problems related to independent sets can be simultaneously addressed using Lex graphs, revealing a unified approach based on sequence parameters.

Contribution

It introduces a novel method showing that sharp upper bounds for independent sets and independence number can be derived through Lex graphs, unifying previously separate results.

Findings

Both extremal bounds depend on parameters of a sequence on Lex graphs

Unified approach simplifies derivation of bounds in extremal problems

Highlights the significance of Lex graphs in extremal combinatorics

Abstract

Extremal problems involving independent sets are much studied. Two of the most important extremal problems in this context are concerned with the sharp upper bounds for the number of independent sets of fixed size and the independence number. In literature, these sharp upper bounds are derived in completely different contexts. In this paper, we show that both of these sharp upper bounds can be derived by considering Lex graphs. More exactly, they depend on two parameters of a sequence defined on them.