Local Properties via Color Energy Graphs and Forbidden Configurations

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 $k$.