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Extremal Combinatorics

Informacje ogólne

Kod przedmiotu: 1000-2M14EC Kod Erasmus / ISCED: 11.3 / (0612) Database and network design and administration
Nazwa przedmiotu: Extremal Combinatorics
Jednostka: Wydział Matematyki, Informatyki i Mechaniki
Grupy: Przedmioty monograficzne dla III - V roku informatyki
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Punkty ECTS i inne: (brak)
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Język prowadzenia: angielski
Rodzaj przedmiotu:

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Skrócony opis:

The purpose of this course is to provide an introduction to the branch of combinatorial theory known as extremal combinatorics. A classical problem in extremal combinatorics is of the following nature: find the minimum or maximum size of a collection of finite objects that satisfies certain conditions and, if possible, characterize structure of the extremal families. We will primarily focus on extremal problems on systems of finite sets. Alongside theorems that are the cornerstones of the theory, there will be special emphasis on major tools and techniques, particularly methods from linear algebra. We will also highlight certain applications to geometry and computer science.

Pełny opis:

1.Sperner systems: Sperner's theorem, the LYM (Lubell-Yamamoto-Meshalkin) inequality, Bollobas' ''set pairs'' theorem.

2.Intersecting set systems: the Erdős-Ko-Rado theorem, Katona's proof using cyclic permutations, Frankl's generalization, the complete intersection theorem of Ahlswede-Khachatrian.

3.Compression techniques and shadows of set systems: the Kruskal-Katona theorem, Lovasz's version, Daykin's proof of Erdős-Ko-Rado using Kruskal-Katona.

4.Sunflowers: The Sunflower Lemma of Erdős-Rado, the Sunflower Conjecture, and an algorithmic application involving the d-HITTING SET problem.

5.The linear algebra bound: the Oddtown theorem and variants, Fisher's inequality, packing complete bipartite graphs (Graham-Pollak).

6.The method of linearly independent polynomials: Covering a cube with hyperplanes, set systems with restricted intersection sizes.

7.Applications to discrete geometry: Two-distance point sets in Euclidian space, a disproof of Borsuk's conjecture using extremal set theory.

8.Exterior products: The exterior algebra of a vector space, a proof of a variant of Bollobas' ''set pairs'' theorem using exterior products. An algorithmic application of Bollobas' theorem to computing representative sets, illustrated using the k-PATH problem.

9.Traces of set systems: VC-dimension of a set system, Sauer-Shelah lemma, applications to computational learning theory.

10.The Four Functions Theorem: Statement and proof, and an application to an extremal problem on union of intersecting families.

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