Mittagsseminar (in cooperation with A. Steger, D. Steurer and B. Sudakov)

Date and Time: Wednesday, May 20, 2009, 12:15 pm

Duration: This information is not available in the database

Location: OAT S15/S16/S17

Speaker: Martin Tancer (Charles Univ., Prague)

Non-representability of finite projective planes by convex sets

It is well known that a finite projective plane cannot be represented by lines in R^d. We prove a similar result when we consider a representation by convex sets (for projective planes that arise from a finite field).

More precisely, we show that for every positive integer d there is a positive integer q_0 with the following property. Let us have a projective plane which arises over GF(q) for q \geq q_0 with the set of lines L. Then there are no convex sets C_l in d-space for lines l from L such that for every l_1,...,l_k the sets C_{l_1},...,C_{l_k} intersect if and only if the lines l_1,...,l_k meet in a point of the projective plane.

If I have enough time I will also explain the main motivation of this problem: a simplicial complex is d-representable if it is the nerve of a collection of convex sets in R^d. A simplicial complex is d-collapsible if it can be reduced to an empty complex by repeatedly removing (collapsing) a face of dimension at most d-1 that is contained in a unique maximal face. Every d-representable simplicial complex is d-collapsible. Alon, Kalai, Matoušek and Meshulam asked whether there is a function f(d) such that every d-collapsible complex is f(d)-representable. Our result on projective planes implies that no such f exists for d > 1.

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