Abstract. Hom (G, H) is a polyhedral complex defined for any two undirected graphs G and H. This construction was introduced by Lovász to give lower bounds for chromatic numbers of graphs. In this paper we initiate the study of the topological properties of this class of complexes. We prove that Hom (Km, Kn) is homotopy equivalent to a wedge of (n−m)dimensional spheres, and provide an enumeration formula for the number of the spheres. As a corollary we prove that if for some graph G, and integers m ≥ 2 and k ≥ −1, we have ̟k 1 (Hom (Km, G)) ̸ = 0, then χ(G) ≥ k + m; here Z2-action is induced by the swapping of two vertices in Km, and ̟1 is the first Stiefel-Whitney class corresponding to this action. Furthermore, we prove that a fold in the first argument of Hom (G, H) induces a homotopy equivalence. It then follows that Hom (F, Kn) is homotopy equivalent to a direct product of (n−2)-dimensional spheres, whileHom (F, Kn) is homotopy equivalent to a wedge of spheres, where F is an arbitrary forest and F is its complement. 1.1. Definition of the main object. 1.

The local chromatic number of a graph was introduced in [12]. It is in between the chromatic and fractional chromatic numbers. This motivates the study of the local chromatic number of graphs for which these quantities are far apart. Such graphs include Kneser graphs, their vertex color-critical subgraphs, the Schrijver (or stable Kneser) graphs; Mycielski graphs, and their generalizations; and Borsuk graphs. We give more or less tight bounds for the local chromatic number of many of these graphs. We use an old topological result of Ky Fan [14] which generalizes the Borsuk-Ulam theorem. It implies the existence of a multicolored copy of the complete bipartite graph K⌈t/2⌉,⌊t/2 ⌋ in every proper coloring of many graphs whose chromatic number t is determined via a topological argument. (This was in particular noted for Kneser graphs by Ky Fan [15].) This yields a lower bound of ⌈t/2 ⌉ + 1 for the local chromatic number of these graphs. We show this bound to be tight or almost tight in many cases. As another consequence of the above we prove that the graphs considered here have equal circular and ordinary chromatic numbers if the latter is even. This partially proves a conjecture of Johnson, Holroyd, and Stahl and was independently attained by F. Meunier

Abstract. In this paper we prove the Lovász Conjecture: If Hom(C2r+1, H) is k-connected, then χ(H) ≥ k + 4, where H is a finite undirected graph, C2r+1 is a cycle with 2r+1 vertices, r, k ∈ Z, r ≥ 1, k ≥ −1, and Hom(G, H) is the cell complex with the vertex set being the set of all graph homomorphisms from G to H, and cells all allowed list H-colorings of G. Our method is to compute, by means of spectral sequences, the obstructions to graph colorings, which lie either directly in the cohomology groups of Hom(C2r+1, Kn), or in the vanishing of the certain powers of Stiefel-Whitney classes of Hom(C2r+1, Kn), viewed as Z2-spaces, resulting in proving even sharper statements.

Abstract. For any two graphs G and H Lovász has defined a cell complex Hom(G, H) having in mind the general program that the algebraic invariants of these complexes should provide obstructions to graph colorings. Here we announce the proof of a conjecture of Lovász concerning these complexes with G a cycle of odd length. More specifically, we show that If Hom(C2r+1, G) is k-connected, then χ(G)≥k + 4. Our actual statement is somewhat sharper, as we find obstructions already in the non-vanishing of powers of certain Stiefel-Whitney classes. 1.

In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs KGr sS, “generalized r-uniform Kneser hypergraphs with intersection multiplicities s. ” It generalized previous lower bounds by Kˇríˇz (1992/2000) for the case s = (1,...,1) without intersection multiplicities, and by Sarkaria (1990) for S = � [n] � k. Here we discuss subtleties and difficulties that arise for intersection multiplicities si> 1: 1. In the presence of intersection multiplicities, there are two different versions of a “Kneser hypergraph,”depending on whether one admits hypergraph edges that are multisets rather than sets. We show that the chromatic numbers are substantially different for the two concepts of hypergraphs. The lower bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset version. 2. The reductions to the case of prime r in the proofs by Sarkaria and by Ziegler work only if the intersection multiplicities are strictly smaller than the largest prime factor of r. Currently we have no valid proof for the lower bound result in the other cases. We also show that all uniform hypergraphs without multiset edges can be represented as generalized Kneser hypergraphs.

Abstract. We use stable Kneser hypergraphs to construct ideals in N which are not nonatomic yet have the Nikod´ym property. 1.

Abstract. There are two possible definitions of the “s-disjoint r-uniform Kneser hypergraph” of a set system T: The hyperedges are either r-sets or r-multisets. We point out that Ziegler’s (combinatorial) lower bound on the chromatic number of an s-disjoint r-uniform Kneser hypergraph only holds if we consider r-multisets as hyperedges. We give a new proof of his result and show by example that a similar result does not hold if one considers r-sets as hyperedges. In case of r-sets as hyperedges and s ≥ 2 the only known lower bounds are obtained from topological invariants of associated simplicial complexes if r is a prime or the power of prime. This is also true for arbitrary r-uniform hypergraphs with r-sets or r-multisets as hyperedges as long as r is a power of a prime. 1.

Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least c () |V (H)| r−1 has bounded chromatic number. This parameter has a long history for graphs (r = 2), and in this paper we begin its systematic study for hypergraphs. ̷Luczak and Thomassé recently proved that the chromatic threshold of near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r + 1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen’s

Abstract. This article is concerned with a general scheme on how to obtain constructive proofs for combinatorial theorems that have topological proofs so far. To this end the combinatorial concept of Tucker-property of a finite group G is introduced and its relation to the topological Borsuk-Ulam-property is discussed. Applications of the Tucker-property in combinatorics are demonstrated. 1.

In Ziegler (2002), the second author presented a lower bound for the chromatic numbers of hypergraphs KGr sS, “generalized r-uniform Kneser hypergraphs with intersection multiplicities s. ” It generalized previous lower bounds by Kˇríˇz (1992/2000) for the case s = (1,...,1) without intersection multiplicities, and by Sarkaria (1990) for S = ([n]) k. Here we discuss subtleties and difficulties that arise for intersection multiplicities si> 1: 1. In the presence of intersection multiplicities, there are two different versions of a “Kneser hypergraph,”depending on whether one admits hypergraph edges that are multisets rather than sets. We show that the chromatic numbers are substantially different for the two concepts of hypergraphs. The lower bounds of Sarkaria (1990) and Ziegler (2002) apply only to the multiset version. 2. The reductions to the case of prime r in the proofs by Sarkaria and by Ziegler work only if the intersection multiplicities are strictly smaller than the largest prime factor of r. Currently we have no valid proof for the lower bound result in the other cases. We also show that all uniform hypergraphs without multiset edges can be represented as generalized Kneser hypergraphs. 1