Plünnecke and Kneser type theorems for dimension estimates

Plünnecke and Kneser type theorems for dimension estimates

Given a division ring K containing the field k in its center and two finite subsets A and B of K∗, we give some analogues of Plünnecke and Kneser Theorems for the dimension of the k-linear span of the Minkowski product AB in terms of the dimensions of the k-linear spans of A and B. We also explain how they imply the corresponding more classical theorems for abelian groups. These Plünnecke type ...

Generalized Kneser Coloring Theorems with Combinatorial Proofs

The Kneser conjecture (1955) was proved by Lovász (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, Matoušek provided the first combinatorial proof of the Kneser conjecture. Here we provide a hypergraph coloring theorem, with a combinatorial proof...

Dimension and Embedding Theorems for Geometric Lattices

Let G be an n-dimensional geometric lattice. Suppose that 1 < e, f < n 1, e + f > n, but e and f are not both n 1. Then, in general, there are E, FE G with dimE=e, dimF=f, EvF=l, and dimEhF=e+f-n-l; any exception can be embedded in an n-dimensional modular geometric lattice M in such a way that joins and dimensions agree in G and M, as do intersections of modular pairs, while each point and lin...

Dimension Estimates for Invariant Setsof

Generalized Kneser Coloring Theorems with Combinatorial Proofs (Erratum)

In [5], we presented a lower bound for the chromatic numbers of hypergraphs KG sS, “generalized r-uniform Kneser hypergraphs with intersection multiplicities s.” It generalized previous lower bounds by Kř́ıž [1, 2] for the case s = (1, . . . , 1) without intersection multiplicities, and by Sarkaria [4] for S = ([n] k ) . The following two problems that arise for intersection multiplicities si > ...