ragde@utcsri.UUCP (04/20/87)
Interest has been expressed in holding a series of informal meetings to study Ramsey theory. The basic idea is that we would decide on a list of topics and take turns presenting the proofs to each other. I think it would be better if only people who are actively involved come to these meetings - i.e., attendance commits you to making a presentation. We would probably use the Graham, Rothschild & Spencer book, plus additional papers as needed. Below is a list of topics from that book that are within the scope of something like this. I'd like people to let me know if they are interested, and what topics specifically they would like to see covered. This wouldn't start until late summer or maybe even early fall. If there is a LOT of interest, it could be made a course, but I hesitate to do this...--PR Full Proof of Ramsey's Theorem The method of Induced Colouring, which is a paradigm for many subsequent proofs. Compactness How infinite versions of Ramsey theorems imply finite versions. Van Der Waerden's Theorem If N is finitely coloured then there exists a colour class containing arbitrarily long arithmetic progressions. Note that it is not possible to find infinite monochromatic arithmetic progressions. Hales-Jewett Theorem If all vectors of length n with components from {1..t} are coloured with r colours, then for n sufficiently large, there exists a monochromatic line - that is, a set of vectors x_1,...,x_n such that the ith component of x_j is either constant as j varies, or equals j. This implies other useful theorems - in particular, VDW's Theorem. Upper and Lower Bounds for Ramsey Numbers Probabilistic methods and explicit constructions; we should also do the Lovasz local lemma, giving conditions under which all of n events occur with nonzero probability (as is needed for existence proofs using the probabilistic method) even if some events are dependent. Induced Ramsey Theorems For any graph G, there exists a graph H such that if the vertices [edges] of G are coloured with r colours, then there exists an induced subgraph of H which is a monochrome G. Restricted Ramsey Theorems We know that any 2-coloured K_6 contains a monochrome K_3. Does there exist a graph G such that any 2-colouring induces a monochrome triangle, and the clique number of G is 3? What about any number of colours, and any size monochrome clique? Canonical Ramsey Theorems No restrictions on the number of colours; what sort of results can we prove? (These have proved useful in lower bounds for parallel machines - an unrestricted colouring is just a function from the input vectors to some range, and these theorems prove that we can restrict input vectors and give such functions useful properties.) Euclidean Ramsey Theorems Suppose we colour every point of R^n with one of two colours. Does there exist a monochrome equilateral triangle of side 1? No if n=2, yes if n=4. Generalize... Graph Ramsey Theory We define R(G,H) to be the smallest number n of vertices such that if we 2-colour a K_n, either there exists a G of colour 1 or an H of colour 2. An example of a theorem is: R(tree with m vertices, K_n) = (m-1)(n-1)+1 Another is: in any 2-colouring of K_5n there exist n disjoint triangles of the same colour; 5n is the best possible result. (n>1). Independence Results Some variations of Ramsey's Theorem can be shown unprovable in Peano arithmetic. Godel's Theorem shows that unprovable statements exist, but this was the first "natural" one. It can be shown both by model-theoretic methods and by lower-bounding the growth rate of associated functions. Infinite Ramsey Theory If all 2-subsets of the natural numbers are 2-coloured, there exists a countably infinite monochrome subset. This is not true for the real numbers if we want an uncountable monochrome subset. What can we say about the reals, and higher cardinals and ordinals? Any cardinal bigger than omega that has the property above must be "inaccessible" - among other things, not reachable by power sets or limits from the usual type of cardinals. Weirdnesses out past infinity... Other topics will be considered as we come across them in journal searches or hear of more recent proofs. --PR