Generalized Kneser coloring theorems with combinatorial proofs

The Kneser conjecture (1955) was proved by Lov\'asz (1978) using the Borsuk-Ulam theorem; all subsequent proofs, extensions and generalizations also relied on Algebraic Topology results, namely the Borsuk-Ulam theorem and its extensions. Only in 2000, Matou\v{s}ek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof, which has as special cases the Kneser conjecture as well as its extensions and generalization by (hyper)graph coloring theorems of Dol'nikov, Alon-Frankl-Lov\'asz, Sarkaria, and Kriz. We also give a combinatorial proof of Schrijver's theorem.Comment: 19 pages, 4 figure

Convex Equipartitions via Equivariant Obstruction Theory

We describe a regular cell complex model for the configuration space F(\R^d,n). Based on this, we use Equivariant Obstruction Theory to prove the prime power case of the conjecture by Nandakumar and Ramana Rao that every polygon can be partitioned into n convex parts of equal area and perimeter.Comment: Revised and improved version with extra explanations, 20 pages, 7 figures, to appear in Israel J. Mat