Localized versions of extremal problems

TL;DR

This paper extends classical extremal combinatorics theorems by replacing global constraints with class-wide inequalities, leading to broader generalizations of key results like Turán's and Erdős-Gallai theorems.

Contribution

It introduces a unified framework for generalizing extremal combinatorics theorems through localized inequalities applicable to entire classes.

Findings

Generalized Turán's theorem under local constraints

Extended Erdős-Gallai theorem for graph sequences

Broadened Erdős-Ko-Rado and Szekeres theorems

Abstract

We generalize several classical theorems in extremal combinatorics by replacing a global constraint with an inequality which holds for all objects in a given class. In particular we obtain generalizations of Tur\'an's theorem, the Erd\H{o}s-Gallai theorem, the LYM-inequality, the Erd\H{o}s-Ko-Rado theorem and the Erd\H{o}s-Szekeres theorem on sequences.