Instructor: Dr. András GYÁRFÁS
Text: The course is based on handouts.
Prerequisite: Familiarity with binomial coefficients may be enough in case of exceptionally talented students but at least one course in
combinatorics or in graph theory is required in general.
A minimum knowledge of linear algebra and discrete probability is advantageous (for Sections 6 and 7).
Course description: The basic concepts of Hypergraph Theory are introduced. Using this framework, classical and relatively new
results are discussed from the theory of finite set systems.
The emphasis is on studying several successful proof methods including the Linear Algebra and the Probability methods.
Here we add the CO2 MENU CARD provided by the chef(instructor) of the course:1. BASIC NOTIONS AND EXAMPLES
1.1. Definitions. (Incidence matrices, Duality.)
1.2. Examples of hypergraphs. (Paths and cycles, Linear spaces).
2. CHROMATIC NUMBER AND GIRTH
2.1. Chromatic number.
2.2. Graphs from the Hall of Fame. (Graphs of Zykov, Mycielski, Tutte, Shift graph, Kneser graph.)
2.3. How to glue hypergraphs to get graphs? (Nesetril-Rödl hypergraph.)
3. A LOOK AT RAMSEY THEORY.
3.1. Ramsey numbers. (Pigeonholes and parties, Existence of Ramseynumbers, Upper bound on R(n), Many colors and non-diagonal
Ramsey numbers, Exact values of Ramsey numbers.)
3.2. Van der Waerden numbers.
3.3. Tic-tac-toe and Hales-Jewett theorem.
4. COUNTING AND PROBABILITY.
4.1. Proofs by counting. (Antichains, intersecting hypergraphs,3-chromatic uniform hypergraphs.)
4.2. Probability method. (The Erdõs lower bound on R(n), Lower bound for W(k), Tournaments, Large transitive subtournaments, Existence versus construction, Paradoxical tournaments, Hamiltonian paths.)
4.3. Local Lemma. (Erdõs-Lovász theorem, Improved lower bound on W(k), Even cycles in regular digraphs.)
4.4. Jokes (Triangle is 2-chromatic, Spencer's injections.)
5. LINEAR ALGEBRA METHODS.
5.1. The dimension bound. (Odd town, Fisher inequality, A cubic lower bound for R(n), Two-distance sets, Cross-intersecting hypergraphs.)
5.2. Homogeneous linear equations. (Partition into complete bipartite graphs, Beck-Fiala theorem.)
5.3. Eigenvalues. (Regular graphs of girth five.)
SPECIAL OFFERS1. FRUIT SALAD.
1.1. Distinct representatives of sets.
1.2. Symmetric chain decomposition.
1.3. A property of n sets on n elements.
1.4. A property of n+1 sets on n elements.
1.5. A property of n+2 sets on n elements.
1.6. 3-color-critical hypergraphs.
1.7. Sunflower theorem.
1.8. Sum-free subsets of numbers.
2. ADVANCED MENU.
2.1. Factorization of hypergraphs - Baranyai theorem.
2.2. Normal hypergraphs and perfect graphs - Lovász theorem.
2.3. Constructive superpolynomial lower bound for Ramsey numbers.
2.4. Hypergraphs in geometry: the fall of Borsuk conjecture.
2.5. Shelah's proof of Hales-Jewett theorem.
2.6. Graphs with large chromatic number and girth - the proof ofErdõs.
2.7. Addressing with the squashed cube - Winkler's theorem.