7
votes
2answers
417 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 ...
7
votes
3answers
308 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 ...
4
votes
2answers
581 views

Partition calculus question

For $m,n,k < \omega$, consider the equation $X \to (\omega \times k)^{m}_{n}$ What is the smallest $X$ known to satisfy it? Baumgartner-Hajnal theorem gives a satisfactory answer for $m=2$, but ...
8
votes
1answer
460 views

monochromatic cycle-free colouring of the complete graph on R?

Hi So there is an edge-colouring of a complete graph on R (the reals), with countably many colours that as no monochromatic triangle. To find it map R to (0,1) write the numbers in binary and if 2 ...