Confirming a conjecture of Gyárfás, we prove that, for all natural numbers k and r , the vertices of every r-edge-coloured complete k-uniform hypergraph can be partitioned into a bounded number (independent of the size of the hypergraph) of monochromatic tight cycles. We further prove that, for all natural numbers p and r , the vertices of every r-edge-coloured complete graph can be partitioned into a bounded number of p-th powers of cycles, settling a problem of Elekes, Soukup, Soukup and Szentmiklóssy. In fact we prove a common generalisation of both theorems which further extends these results to all host hypergraphs of bounded independence number.
Given a metric space M that contains at least two points, the chromatic number χ (R n ∞ , M) is defined as the minimum number of colours needed to colour all points of an n−dimensional space R n ∞ with the max-norm such that no isometric copy of M is monochromatic. The last two authors have recently shown that the value χ (R n ∞ , M) grows exponentially for all finite M. In the present paper we refine this result by giving the exact value χ M such that χ (R n ∞ , M) = (χ M + o(1)) n for all 'one-dimensional' M and for some of their Cartesian products. We also study this question for infinite M. In particular, we construct an infinite M such that the chromatic number χ (R n ∞ , M) tends to infinity as n → ∞.
The following generalisation of the Erdős unit distance problem was recently suggested by Palsson, Senger and Sheffer. Given k positive real numbers δ 1 , . . . , δ k , a, where the maximum is taken over all δ? Improving the results of Palsson, Senger and Sheffer, we essentially determine this maximum for all k in the planar case. error term It is only for k ≡ 1 (mod) 3 that the answer depends on the maximum number of unit distances in a set of n points. We also obtain almost sharp results for even k in 3 dimension.
We consider four problems. Rogers proved that for any convex body K, we can cover R d by translates of K of density very roughly d ln d. First, we extend this result by showing that, if we are given a family of positive homothets of K of infinite total volume, then we can find appropriate translation vectors for each given homothet to cover R d with the same (or, in certain cases, smaller) density.Second, we extend Rogers' result to multiple coverings of space by translates of a convex body: we give a non-trivial upper bound on the density of the most economical covering where each point is covered by at least a certain number of translates.Third, we show that for any sufficiently large n, the sphere S 2 can be covered by n strips of width 20n/ ln n, where no point is covered too many times.Finally, we give another proof of the previous result based on a combinatorial observation: an extension of the Epsilon-net Theorem of Haussler and Welzl. We show that for a hypergraph of bounded Vapnik-Chervonenkis dimension, in which each edge is of a certain measure, there is a not-too large transversal set which does not intersect any edge too many times.
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