Lexicographic Ramsey theory

Lexicographic Ramsey Theory

Given positive integers d and n, there is an integer N such that for every injective map ffrom { 1 ..... N} a into R there is a subset A = A1 x A2 x ... × Ad of { 1 ..... N} a such that (1) each Aj has n elements, (2) the restriction of f to A is monotone in each coordinate, (3) there is an ordering of the coordinates such that f on A is texicographic with respect to that ordering. Because inje...

Ramsey vs. Lexicographic Termination Proving

Termination proving has traditionally been based on the search for (possibly lexicographic) ranking functions. In recent years, however, the discovery of termination proof techniques based on Ramsey’s theorem have led to new automation strategies, e.g. size-change, or iterative reductions from termination to safety. In this paper we revisit the decision to use Ramsey-based termination arguments...

Ramsey Theory

Ramsey Theory

Ramsey Theory is the study of inevitable substructures in large (usually discrete) objects. For example, consider colouring the edges of the complete graph Kn with two colours. In 1930, Ramsey [13] proved that if n is large enough, then we can find either a red complete subgraph on k vertices or a blue complete subgraph on ` vertices. We write Rpk, `q for the smallest such n. Another famous exa...

Ramsey Theory

We give a proof to arithmetic Ramsey’s Theorem. In addition, we show the proofs for Schur’s Theorem, the Hales-Jewett Theorem, Van der Waerden’s Theorem and Rado’s Theorem, which are all extensions of the classical Ramsey’s Theorem.