You are invited to our Monday Lecture & Colloquium on Dec 17th at 14:15 h & 16:00 h at FU Berlin. Lecture: Monday, Dec 17th - 14:15 h =============================================== Shagnik Das (Freie Universität Berlin): Randomly perturbed Ramsey problems =============================================== Abstract: The combinatorial properties much-loved Erdős-Rényi random graph G(n,p), which has n vertices and whose edges are present independently with probability p, have been comprehensively studied in the decades since its introduction. In recent years, much research has been devoted to the randomly perturbed graph model, introduced in 2003 by Bohman, Frieze and Martin. In this talk we shall present and motivate this new model of random graphs, and then focus on the Ramsey properties of these randomly perturbed graphs. More precisely, given a pair of graphs (F,H), we ask how many random edges must be added to a dense graph G to ensure that any two-colouring of the edges of the perturbed graph has either a red copy of F or a blue copy of G. This question was first studied in 2006 by Krivelevich, Sudakov and Tetali, who answered it in the case of F being a triangle and H being a clique. We generalise these results, considering pairs of larger cliques, and, should the audience be willing (but even otherwise), shall take a quick look at some of the ideas required in our proofs. This is joint work with Andrew Treglown (Birmingham). Coffee & Tea Break Colloquium: 16:00 h =============================================================== Marie Brandenburg (Freie Universität Berlin): Product-Mix Auctions, Competitive Equilibrium and Lattice Polytopes =============================================================== Abstract: In 2007, when the credit crisis began, the Bank of England approached the economist Paul Klemperer with the need of a new auction design that would enable them to give liquidity to all banks in an economically efficient way. Since then, Klemperer’s Product-Mix Auction design became more and more relevant, with particular interest in when competitive equilibrium exists, that is, when a set of (indivisible) goods can be split such that exactly one bid of each bidder is fulfilled. Originally being a question of economics, this can be translated into a question of specific lattice polytopes that rely on underlying graphs. I will present the Product-Mix Auction setting, the translation to polytopes and first results on when competitive equilibrium exists. Location: Room 005 - Ground Floor Freie Universität Berlin Takustr. 9 14195 Berlin