The open problems project

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You can find the individual projects here and the open positions here. Their origins go back to when Eszter Klein asked the following question: Is it true that for any integer k there is a smallest integer g k such that any set of at least g k points in the plane contains a convex subset of cardinality k?

Prominent variations of the original question from Eszter Klein are to require empty convex sets and to ask about the number of convex empty subsets of a certain size. For many questions in discrete geometry the goal is to obtain extremal configurations, that is, structures attaining the minimum or maximum number of the considered geometric objects.

Informally, Horton sets are very anti-convex, and thus it is natural to study the behavior of sets of different degrees of convexity. One proposed approach in this respect is to investigate the size and the number of convex subsets in k-convex point sets.

In this project we investigate questions of the above flavor, for example the search for convex holes in k-convex point sets and the number of general that is, not necessarily convex k-holes in point sets. These questions are notoriously difficult though important, so that even partial solutions are expected to have impact. The Doctoral Program "Discrete Mathematics" offers an advanced PhD training and research program which is run jointly by. You can find the individual projects here and the open positions here. Their origins go back to when Eszter Klein asked the following question: Is it true that for any integer k there is a smallest integer g k such that any set of at least g k points in the plane contains a convex subset of cardinality k?

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Prominent variations of the original question from Eszter Klein are to require empty convex sets and to ask about the number of convex empty subsets of a certain size. For many questions in discrete geometry the goal is to obtain extremal configurations, that is, structures attaining the minimum or maximum number of the considered geometric objects. Informally, Horton sets are very anti-convex, and thus it is natural to study the behavior of sets of different degrees of convexity.

Classic texts on unsolved problems in various areas of mathematics are Croft et al.


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Clay Mathematics Institute. Croft, H. Unsolved Problems in Geometry. New York: Springer-Verlag, p.

Doctoral Program "Discrete Mathematics"

Demaine, E. Emden-Weinert, T. Eppstein, D. Finch, S. Guy, R. Unsolved Problems in Number Theory, 3rd ed. New York: Springer-Verlag, Kimberling, C. Klee, V. Meschkowski, H.

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Unsolved and Unsolvable Problems in Geometry. Ogilvy, C.

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Tomorrow's Math: Unsolved Problems for the Amateur, 2nd ed. New York: Oxford University Press, New York: Dover, pp.

[PDF] The Open Problems Project - Semantic Scholar

Ramachandra, K. Indian Nat. Part A 64 , ,