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The aim of this note is to give an elementary proof of the following fact: given 3 red convex sets and 3 blue convex sets in $\mathbb{E}^3$, such that every red intersects every blue, there is a line transversal to the reds or there is a line transversal to the blues. This is a special case of a theorem of Montajano and Karasev \cite{MK} and generalizes, in a sense, the colourful Helly theorem due to Lov\'asz (cf. \cite{BL})., Comment: To appear in Annals of Combinatorics

We introduce classes of edge-colourings of the complete graph -- that we call nice and beautiful -- and study how many heterochromatic spanning trees appear under such colourings. We prove that if the colouring is nice, there is at least a quadratic number of different heterochromatic trees; and if the colouring is beautiful there is an exponential number of different such trees., Comment: Typos corrected, figures added

We prove that, if every cycle of $D$ is an $H$-cycle, then $D$ has an $H$-kernel by walks.

Given D and H two digraphs, D is H-coloured iff the arcs of D are coloured with the vertices of H. After defining what do we mean by an H-walk in the coloured D, we characterise those H, which we call panchromatic patterns, for which all D and all H-colourings of D admit a kernel by H-walks. This solves a problem of Arpin and Linek from 2007.

In 2003 Daniel K. Biss published, in the Annals of Mathematics, what he thought was a solution of a long standing problem culminating a discovery by Gelfand and MacPherson. Six years later he was encouraged to publish an "erratum" of his prove, observed by Nikolai Mnev; up to now, the homotopy type of the MacPhersonian had remained a mistery... The aim of this lecture is to convince the attendee of the fact that, using a compleatly different aproach to those used before, we can prove that the acyclic MacPhersonian has the homotopy type of the affine Grassmannian.

We characterize which systems of sign vectors are the cocircuits of an oriented matroid in terms of the cocircuit graph., Comment: 6 pages

We present an algorithm which takes a graph as input and decides in cubic time if the graph is the cocircuit graph of a uniform oriented matroid. In the affirmative case the algorithm returns the set of signed cocircuits of the oriented matroid. This improves an algorithm proposed by Babson, Finschi and Fukuda. Moreover we strengthen a result of Montellano-Ballesteros and Strausz about crabbed connectivity of cocircuit graphs of uniform oriented matroids., Comment: 9 pages