0
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0answers
33 views

Some small (and probably easy) implication from combinatorics paper

I'm reading https://people.math.osu.edu/bergelson.1/PolSz.pdf . Question is about part from (1.7) on page 16. Why $ \lambda(\chi_{1})=\lambda(\chi_{1}) $? There's no problem if in 4-th verse ...
1
vote
2answers
88 views

Reconciling two statements of Ramsey's Theorem.

In the appendix to Shelah's book on Classification Theory: THEOREM 2.1 (Ramsey's Theorem): (1) For any infinite ordered set $I$, and $n$-place function $f$ from $I$, with range of cardinality ...
11
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5answers
792 views

Why König's lemma isn't “obvious”?

I keep facing K├Ânig's lemma "Every finitely branching infinite tree over $\mathbb{N}$ has infinite branch". Why it is not taken "obvious" but needs a careful proof? It seems somewhat obvious, but I ...
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1answer
71 views

Ramsey's theorem.

For $k=2$, from this post A "geometrical" representation for Ramsey's theorem, how one can deduce the theorem from Constant $f:[\mathbb{N}]^2\to \{1,2\}$ (part 2), or by knowing that ...
10
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1answer
98 views

Coloring of positive integers

Suppose $f:\mathbb{Z}^+\longrightarrow X$ is a function, with $X$ a finite set. Is it true that there are $a,b\in\mathbb{Z}^+$ such that $f(a)=f(b)=f(a+b)$.