Let q = 2 be, for some ∈ N, and let n = q 2 +q +1. By exhibiting a complete coloring of the edges of K n , we show that the pseudoachromatic number (G n ) of the complete line graph G n = L(K n )-or the pseudoachromatic index of K n , if you will-is at least q 3 +q. This bound improves the implicit bound of Jamison [Discrete Math 74 (1989), 99-115] which is given in terms of the achromatic number:We also calculate, precisely, the pseudoachromatic number when q +1

We will prove the following generalisation of Tverberg's Theorem: given a set S ⊂ R d of (r + 1)(k − 1)(d + 1) + 1 points, there is a partition of S in k sets A 1 , A 2 , . . . , A k such that for any C ⊂ S of at most r points, the convex hulls of A 1 \C, A 2 \C, . . . , A k \C are intersecting. This was conjectured first by Natalia García-Colín (Ph.D. thesis, University College of London, 2007).

Let ψ 1 (K n ) and α 1 (K n ) be the pseudoachromatic index and the achromatic index of the complete graph, respectively. Let γ ≥ 2 be a positive integer q = 2 γ and m = (q + 1) 2 . In this paper exhibiting three closely related edge-colorings of the complete graph we prove that for a ∈ {0, 1, 2}:

In this paper we study the topology of transversals to a family of convex sets as a subset of a Grassmanian manifold. This topology seems to be ruled by a combinatorial structure which we call a separoid. With these combinatorial objects and the topological notion of virtual transversal we prove a Borsuk-Ulam-type theorem which has as a corollary a generalization of Hadwiger's theorem.

Let Π q be the projective plane of order q, let ψ(m) := ψ(L(K m )) the pseudoachromatic number of the complete line graph of order m, let a ∈ {3, 4, . . . , q 2 + 1} and m a = (q + 1) 2 − a. In this paper, we improve the upper bound of ψ(m) given by Araujo-Pardo et al. [J Graph Theory 66 (2011), 89-97] and Jamison [Discrete Math. 74 (1989), 99-115] in the following values: if x ≥ 2 is an integer and m ∈ {4x 2 − x, . . . , 4x 2 + 3x − 3} then ψ(m) ≤ 2x(m − x − 1).On the other hand, if q is even and there exists Π q we give a complete edge-colouring of K ma with (m a − a)q colours. Moreover, using this colouring we extend the previous results for a =