Local Properties via Color Energy Graphs and Forbidden Configurations

TL;DR

This paper advances the understanding of the local properties problem in combinatorics by developing higher color energy techniques and applying extremal graph theory to derive new bounds for various problem variants.

Contribution

It introduces higher color energy concepts and combines them with extremal graph theory to improve bounds in the local properties problem and related Ramsey and distances problems.

Findings

Derived new bounds for the local properties problem.

Generalized color energy to higher levels.

Connected color energy with extremal graph theory results.

Abstract

The local properties problem of Erd\H{o}s and Shelah generalizes many Ramsey problems and some distinct distances problems. In this work, we derive a variety of new bounds for the local properties problem and its variants. We do this by continuing to develop the color energy technique --- a variant of the concept of additive energy from Additive Combinatorics. In particular, we generalize the concept of color energy to higher color energies, and combine these with Extremal Graph Theory results about graphs with no cycles or subdivisions of size $k$.