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I am reading Jech's Set Theory, in particular the chapter about large cardinals. After discussing measurable cardinals he moves on to weakly-compact cardinals, which have been discussed far earlier in the book.
I went back to the chapter dealing with weakly-compact cardinals and began retracing the definition.

Eventually, it came into this:

We denote $[k]^n = \{X \subset \kappa | |X| = n\}$. If $\lambda$ is a cardinal, we denote $\kappa \to (\lambda)^2$ when for every partition of $[\kappa]^2$ into $2$ we have $H \subset \kappa$ that is of cardinality $\lambda$, and for which $[H]^2$ is strictly in one part.

And we say that $\kappa$ is weakly-compact if it satisfies the property $\kappa \to (\kappa)^2$.

The problem is that I'm a bit lost in all those definitions, and not even sure about the $\kappa \to (\lambda)^2$ notation.

My questions are, if so, can someone help me make some sense into those definitions, and is there an equivalent definition for weakly-compact cardinals which can help me understand their properties better?

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2 Answers 2

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There are many ways to think of these definitions. Here's a way one can understand for example what $[\kappa]^2$ means and what it means to have a homogeneous subset, that I find intuitive.

Assume that you have a complete undirected graph with $\kappa$ many nodes. That means that you have $\kappa$ points and you connect every pair of them with a line. Now assume that we have two colours, red and blue, and that every line between two nodes is coloured either red or blue. A subset of these $\kappa$ nodes is called homogeneous if the lines between each and every of its nodes have the same colour (that's the same as saying that it has a complete subgraph whose lines are of one colour only).

Now we say that $\kappa\to(\lambda)^2$ is true if no matter how we paint those lines using two colours, we can find a homogeneous set of cardinality $\lambda$. That is, for every way we colour the lines we can find $\lambda$ many nodes that every line between them is of the same colour.

In other words, for every function $f:[\kappa]^2\to 2$ (you can think of this as a function that sends every two elements of $\kappa$ to one of the 2 colours) we can find a subset of $\kappa$ (let's call it$H$) that has cardinality $\lambda$ such that for every $x,y,z,w\in H$ we have $f({ x,y} )=f({ z,w} )$.

To generalize this if for every function $f:[\kappa]^n\to\mu$ (again you can see this as a function that sends every $n$ elements of $\kappa$ to one of the $\mu$ colours, or that it partitions the subsets of $\kappa$ of cardinality $n$ into $\mu$ partitions) we can find a set $H$ such that $\left|H\right|=\lambda$ and for every $x_1,\ldots,x_n,y_1,\ldots,y_n\in H$ we have $f({ x_1,\ldots,x_n} )=f({ y_1,\ldots,y_n} )$ then we say that $\kappa\to(\lambda)^n_\mu$ is true.

The arrow notation, even though appears weird at first is used because the property remains true if we replace the cardinal in the left side of the arrow with a larger cardinal or if we replace any cardinal in the right side of the arrow with a smaller cardinal (if the subscript in the left side of the arrow is omitted then it is assumed that it is 2). It should be obvious that the notation only has meaning if $\lambda<\kappa$.

There are many equivalent definitions of weakly compact cardinals. Jech's book mentions most of them, but here are some. If $\kappa$ is a cardinal then it is weakly compact iff:

$\kappa\to(\kappa)^2$ is true

$\kappa\to(\kappa)^n_\lambda$ where $\lambda<\kappa$ is true

$\kappa$ is inaccessible and has the tree property, that is, if you have a tree with height $\kappa$ and each level of the tree has less than $\kappa$ elements then it has a branch of length $\kappa$

$\kappa$ is $\prod^1_1\textrm{-indescribable}$.

$\kappa$ is inaccessible and the languages $\mathcal{L}_{\kappa,\omega}$ satisfy the weak compactness theorem (hence the name). This means that in a language where we allow $\lambda$ conjunctions and disjunctions (where $\lambda<\kappa$), if we have a set of sentences $\Sigma$ where$\left|\Sigma\right|=\kappa$ and every subset of $\Sigma$ with less than $\kappa$ sentences is satisfiable then $\Sigma$ is satisfiable.

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Let me add to Ivan's answer another equivalent characterization of weak compactness, which might appeal to you, as it makes them resemble miniature measurable cardinals.

Namely, if $\kappa$ is a cardinal and $\kappa^{<\kappa}=\kappa$, then $\kappa$ is weakly compact if and only if for every transitive set $M$ of size $\kappa$ with $\kappa\in M$, there is another transitive set $N$ and an elementary embedding $j:M\to N$ having critical point $\kappa$.

This embedding characterization admits myriad forms. For example, one can insist that $M\models ZF^-$ or even $M\models ZFC$, and that $M^{<\kappa}\subset M$, or that every $A\subset \kappa$ can be placed into such an $M$, and so on. One can even insist that $j\in N$, a property known as the Hauser property.

These various embedding formulations of weak compactness allow one to borrow many of the methods and techniques from much larger large cardinals, which are most often described in terms of embeddings, and apply them with weakly compact cardinals. For example, using Easton support forcing iterations, one can control the value of $2^\kappa$ while preserving the weak compactness of $\kappa$.

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Hello professor Hamkins. The property you mention in the second paragraph is very interesting, could you please tell me where I can find a proof of it? –  Camilo Arosemena May 1 at 3:41

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