Summary of Papers

1. A new approach to the chromatic number of the square of Kneser graph K(2k + 1, k), submitted. Füredi in early 2000 proposed to determine the chromatic number χ of the square of the Kneser graph K(n, k). The problem is interesting when 2k + 1 ≤ n ≤ 3k − 2. The problem has turned out to be surprisingly difficult, and is still open even in the first case n = 2k + 1. It is believed to be χ = 2k + c for n = 2k + 1 where c is a constant. The best known upper bound is 8/3k + 20/3 by Kim and Park in 2014. In this paper, we provide improved upper bounds of 5/2k + c, where c is a constant in {1/2, 5/2, 4, 5}, depending on k ≥ 2, by employing graph homomorphisms, cartesian products of graphs, and linear congruences in number theory integrated into combinatorial arguments. 2. Rectilinear equilateral sets in Rn, Geombinatorics, 20 (2011) 151–167. Kusner (1983) conjectured that the maximum size of a set whose elements are pairwise equidistant under the l1-norm in Rn is 2n. If true, this would be sharp. The conjecture has been proved for n = 3 (in 1998) and n = 4 (in 2000). Recently, Alon and Pudlak gave an upper bound of O(n lnn). These proofs are very involved and use ideas from embeddings of metric spaces, approximation theory, etc. However, in the words of the authors, their methods won’t solve the full conjecture. We show that the Kusner conjecture is true if there exists an equidistant set of size 3n/2 on the surface of the generalized-octahedron with radius 1 in Rn. Related to this, we show that the maximum size of an equidistant set on the surface of the unit generalized-octahedron without its extreme point is at most n, and this is sharp. Also, we show that if F is a family of the unit generalized-octahedrons such that every two members of F intersect, then every three members of F must intersect. We are working on extending this result to showing that a same F with at least 2n+1 members must be an intersecting family. This would be enough to prove the Kusner conjecture.