Abstracts for Part III Seminars Lent 2008 - Combinatorics
Refer to the overview page for room numbers and details of other talks in the seminar series.
12:05 - Nicholas Korpelainen: Do We Really Need Density?
This talk will relate to the essay topic 'Sparse Ramsey Theory'. It will be accessible to anyone who is familiar with the basic definitions of graph theory.
We'll illustrate, through some beautiful examples, an inductive technique called 'Nesetril-Rödl Amalgamation'. This powerful tool allows us to build sparse structures with substructures that we'd typically expect to crop up in dense cases!
16:05 - Matas Šileikis: Some Hypergraphs are Stable while Being Almost Extremal
This is a completely self-contained introduction to Turán-type problems inspired by the essay topic "Extremal Hypergraph Theory". It will lead to so called stability results.
A result of this kind for a certain 3-graph (called the Fano plane) will be stated and applied to deduce the maximum number of hyperedges of a Fano-free 3-graph.
17:00 - Sean Lip: How to Find the Winning Move: An Introduction to Combinatorial Game Theory
The subject matter of this talk does not relate to any Part III course or essay. The purpose is to present basic methods of analysing two-player impartial combinatorial games, leading up to the rather nice result that any such game (subject to a couple of other conditions) can be reduced to Nim. This is useful because Nim has been completely solved, and its solution isn't hard.
Virtually no prerequisites are assumed; anything needed will be covered in the talk, and it should certainly be accessible to undergraduates. Many examples will be provided.