The Chromatic Number of Two Families of Generalized Kneser Graphs Related to Finite Generalized Quadrangles and Finite Projective 3-Spaces

Let $\Gamma$ be the graph whose vertices are the chambers of the finite projective space $\mathrm{PG}(3,q)$ with two vertices being adjacent when the corresponding chambers are in general position. It is known that the independence number of this graph is $(q^2+q+1)(q+1)^2$. For $q\geqslant 43$ we determine the largest independent set of $\Gamma$ and show that every maximal independent set that is not a largest one has at most constant times $q^3$ elements. For $q\geqslant 47$, this information is then used to show that $\Gamma$ has chromatic number $q^2+q$. Furthermore, for many families of generalized quadrangles we prove similar results for the graph that is built in the same way on the chambers of the generalized quadrangle.

Query Complexity of Sampling and Small Geometric Partitions

In this paper we study the following problem.Discrete partitioning problem (DPP). Let$\mathbb{F}_q$Pndenote then-dimensional finite projective space over$\mathbb{F}_q$. For positive integerk⩽n, let {Ai}i= 1Nbe a partition of ($\mathbb{F}_q$Pn)ksuch that:(1)for alli⩽N,Ai= ∏j=1kAji(partition into product sets),(2)for alli⩽N, there is a (k− 1)-dimensional subspaceLi⊆$\mathbb{F}_q$Pnsuch thatAi⊆ (Li)k.What is the minimum value ofNas a function ofq, n, k? We will be mainly interested in the casek=n.DPP arises in an approach that we propose for proving lower bounds for the query complexity of generating random points from convex bodies. It is also related to other partitioning problems in combinatorics and complexity theory. We conjecture an asymptotically optimal partition for DPP and show that it is optimal in two cases: when the dimension is low (k=n= 2) and when the factors of the parts are structured, namely factors of a part are close to being a subspace. These structured partitions arise naturally as partitions induced by query algorithms. Our problem does not seem to be directly amenable to previous techniques for partitioning lower bounds such as rank arguments, although rank arguments do lie at the core of our techniques.