A tower lower bound for the degree relaxation of the Regularity Lemma

TL;DR

This paper establishes a lower bound on the number of clusters needed for $ ext{ extonehalf}$-degular partitions, showing that even relaxed regularity conditions do not significantly reduce the complexity bounds of Szemerédi's regularity lemma.

Contribution

The paper proves that any $ ext{ extonehalf}$-degular partition can require a tower-type number of clusters, demonstrating the limitations of relaxing regularity conditions.

Findings

Existence of graphs requiring tower($ heta( ext{ extonehalf} ext{-degular}))$ clusters for $ ext{ extonehalf}$-degular partitions.

Degularity relaxation does not substantially improve the bounds of Szemerédi's regularity lemma.

Lower bounds match the complexity of regular partitions, indicating inherent limitations.

Abstract

It is well-known that if $(A,B)$ is an $\tfrac{\varepsilon}{2}$-regular pair (in the sense of Szemer\'edi) then there exist sets $A'\subset A$ and $B'\subset B'$ with $|A'|\leq \varepsilon|A|$ and $|B'|\leq \varepsilon|B|$ so that the degrees of all vertices in $A\setminus A'$ differ by at most $\varepsilon|B|$ and the degrees of all vertices in $B\setminus B'$ differ by at most $\varepsilon|A|$. We call such a property "$\varepsilon$-degularity". This leads to the notion of an "$\varepsilon$-degular" partition of a graph in the same way as the definition of $\varepsilon$-regular pairs leads to the notion of $\varepsilon$-regular partitions. We show that there exist graphs in which any $\varepsilon$-degular partition requires the number of clusters to be $\mathrm{tower}(\Theta(\varepsilon^{-1/3}))$. That is, even though degularity is a substantial relaxation of regularity, in general one cannot improve much on the bounds that come with Szemer\'edi's regularity lemma.