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Category Archives: Open problems
Peter Mani (a photograph by Emo Welzl) Simple polytopes, puzzles Micha A. Perles conjectured in the ’70s that the graph of a simple -polytope determines the entire combinatorial structure of the polytope. This conjecture was proved in 1987 by Blind … Continue reading
Laci and Kati This is the first of a few posts which are spin-offs of the extremal combinatorics series, especially of part III. Here we talk about Lovasz’s geometric two families theorem. 1. Lovasz’s two families theorem Here … Continue reading
Imre Barany, Rade Zivaljevic, Helge Tverberg, and Sinisa Vrecica Recall the beautiful theorem of Tverberg: (We devoted two posts (I, II) to its background and proof.) Tverberg Theorem (1965): Let be points in , . Then there is a partition of … Continue reading
George Dantzig and Leonid Khachyan In this part we will not progress on the diameter problem that we discussed in the earlier posts but will rather describe a closely related problem for directed graphs associated with ordered families of sets. The role models for … Continue reading
Compression We describe now a nice proof technique called “shifting” or “compression” and mention a few more problems. The Sauer-Shelah Lemma: Let . Recall that a family shatters a set if for every there is such that … Continue reading
. Shattering Let us return to extremal problems for families of sets and describe several basic theorems and basic open problems. In the next part we will discuss a nice proof technique called “shifting” or “compression.” The Sauer-Shelah (-Perles -Vapnik-Chervonenkis) Lemma: (Here we write .) … Continue reading
6. First subexponential bounds. Proposition 1: How to prove it: This is easy to prove: Given two sets and in our family , we first find a path of the form where, and . We let with and consider the family … Continue reading
Let us consider another strategy to deal with our diameter problem. Let us try to associate other graphs to our family of sets. Recall that we consider a family of subsets of size of the set . Let us now associate … Continue reading
3. What we will do in this post and and in future posts We will now try all sorts of ideas to give good upper bounds for the abstract diameter problem that we described. As we explained, such bounds apply … Continue reading