# Permutation Groups, Analysis of Boolean Functions, Finite Geometry, Coding Theory and Algebraic Graph Theory

Important mathematical concepts get reinvented many times. In my recent work with Yuval Filmus we explored objects that are called (in random ordering) Boolean degree 1 functions, Cameron-Liebler line classes, equitable partitions, completely regular strength 0 codes with covering radius 1, intriguing sets, perfect colorings, sets with dual width 1, tactical decompositions or tight sets — all depending on the context and who you ask. While the article with Yuval explains to some extent how these notions connect, a research article does not seem to be the right format to explain all concepts in sufficient detail. This post tries to amend this. It also prepares a future post which will elaborate my research with Yuval in more detail. (more…)

Let $\Gamma$ be a graph with a vertex set $V$ and an adjacency relation $\sim$. A coloring of $\Gamma$ is a map from $V$ from $V$ to a set of colors $C$ such that no pair of adjacent vertices gets mapped to the same element of $C$. The chromatic number of $\Gamma$ is the smallest number of colors needed to color $\Gamma$. One of the nicest graphs to investigate is the Kneser graph: Here $V$ consists of all subsets of $\{ 1, 2, \ldots, n \}$ of size $k$ and two vertices are adjacent if the corresponding $k$-sets are disjoint. For $k=2$ and $n=5$, this is the famous Petersen graph: