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 pro...

We present a necessary and sufficient condition for a graph of odd-girth 2k + 1 to bound the class of K4-minor-free graphs of odd-girth (at least) 2k + 1, that is, to admit a homomorphism from any such K4-minor-free graph. This yields a polynomial-time algorithm to recognize such bounds. Using this condition, we first prove that every K4-minor free graph of odd-girth 2k+1 admits a homomorphism ...

This note contains three versions of a result which is sometimes referred to as Kneser’s lemma and states that (with respect to a fixed triangulation) there is a bound on the number of disjointly embedded normal surfaces in a 3–manifold which satisfy an extra non–triviality condition. One often stipulates that no two of the surfaces be parallel in M . We relax this condition and merely assume t...

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 simila...

At every point, a smooth plane curve can be approximated, to second order, by circle; this circle is called osculating. One may think of the osculating as passing through three infinitesimally close points curve. A vertex point at which hyper-osculates: it approximates third order. Equivalently, critical curvature function. Consider (necessarily non-closed) curve, free from vertices. The classi...

In Ziegler (2002), the second author 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 (1992/2000) for the case s = (1, . . . , 1) without intersection multiplicities, and by Sarkaria (1990) for S = ([n] k ) . Here we discuss subtleties and difficulti...