4
votes
1answer
178 views

Is Van der Waerden's function elementary

Van der Waerden's function was proved to have elementary upper bound on growth rate. Is the Van der Waerden's function itself elementary in the sense of Kalmar?
3
votes
1answer
248 views

Geometric van der waerden theorem

Van der Waerden theorem states that sufficiently long initial segment of the natural numbers when divided into $r$ parts contains an arithmetic progression of length $k$. The length of the initial ...
3
votes
1answer
102 views

Multipartite Ramsey theorem

Given $c<\infty$ colors, positive integers $k_1,\dots,k_n$ and positive integers $N_1,\dots,N_n$. Then there exist positive integers $M_1,\dots,M_n$ so that for disjoint finite sets $A_1,\dots,A_n$ ...
7
votes
2answers
403 views

Deriving Konig's Lemma directly from Infinite Ramsey's Theorem for triples

Let KL denote König's Lemma (for trees over $\mathbb{N}$), and RT(3) denote the Infinite Ramsey Theorem for triples over $\mathbb{N}$ (notation as in Simpson's book Subsystems of second order ...
9
votes
2answers
345 views

From very many sets of fixed measure in a probability space, can we select many that have a positive intersection?

I assume the following Lemma is either well known or, more probably, a Corollary of a much stronger well known Theorem, and I would be grateful for a reference: For all $\delta\in (0,1)$ and all ...
5
votes
0answers
170 views

Van der Waerden like theorem

I am trying to develop bounds for the function B(k) where B(k) is defined as the least such positive integer so that whenever the set $\{1,2,\cdots B(k)\}$ is partitioned into two parts at least one ...
7
votes
3answers
303 views

Extracting countable chains from linear orders

There is a well-known fact in infinite combinatorics asserting that for each infinite linear order $P$ there is a countable subset $R\subseteq P$ of order type either $\omega$ or $\omega^{*}$ (by ...
14
votes
3answers
458 views

Ergodic limits along subsets of $\mathbb{N}.$

Let say that an infinite subsets $A$ of $\mathbb{N}$ is "nice w.r.to ergodic limits", if it can replace $\mathbb{N}$ in the individual ergodic theorem, that is, if it is such that the following ...