On van der Waerden's theorem and the theorem of Paris and Harrington

The Paris-Harrington Theorem

In Ramsey theory, very large numbers and fast-growing functions are more of a rule than an exception. The classical Ramsey numbers R(n,m) are known to be of exponential size: the original proof directly gives the upper bound R(n,m) ≤ ( m+n−2 n−1 ) , and an exponential lower bound is also known. For the van der Waerden numbers, the original proof produced upper bounds that were not even primitiv...

Informal notes on van der Waerden’s Theorem and Ramsey’s Theorem

On van der Waerden’s Theorem. Szemerédi’s Theorem is hard to prove, we rather consider the following weakening about monochromatic arithmetic progressions in two-colored integers. Is it unavoidable to have a monochromatic (m.c.) arithmetic progression of length 3 (a 3-AP) if we two-color the integers? YES, of course. Roth’s Theorem says that the larger of the two color-classes, the one whose de...

Relationship between Kanamori-McAloon Principle and Paris-Harrington Theorem

We give a combinatorial proof of a tight relationship between the Kanamori-McAloon principle and the Paris-Harrington theorem with a number-theoretic parameter function. We show that the provability of the parametrised version of the Kanamori-McAloon principle can exactly correspond to the relationship between Peano Arithmetic and the ordinal ε0 which stands for the proof-theoretic strength of ...

On Applications of Van Der Waerden's Theorem

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Van der Waerden’s Theorem: Variants and “Applications”