weakly compact cardinals and the tree property
A cardinal is weakly compact if and only if it is inaccessible and has the tree property.
Weak compactness implies tree property
Let $\kappa $ be a weakly compact cardinal and let $$ be a $\kappa $ tree with all levels smaller than $\kappa $. We define a theory in ${L}_{\kappa ,\kappa}$ with for each $x\in T$, a constant ${c}_{x}$, and a single unary relation^{} $B$. Then our theory $\mathrm{\Delta}$ consists of the sentences^{}:

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$\mathrm{\neg}\left[B({c}_{x})\wedge B({c}_{y})\right]$ for every incompatible $x,y\in T$

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${\bigvee}_{x\in T(\alpha )}B({c}_{x})$ for each $$
It should be clear that $B$ represents membership in a cofinal branch, since the first class of sentences asserts that no incompatible elements are both in $B$ while the second class states that the branch intersects every level.
Clearly $\mathrm{\Delta}=\kappa $, since there are $\kappa $ elements in $T$, and hence fewer than $\kappa \cdot \kappa =\kappa $ sentences in the first group, and of course there are $\kappa $ levels and therefore $\kappa $ sentences in the second group.
Now consider any $\mathrm{\Sigma}\subseteq \mathrm{\Delta}$ with $$. Fewer than $\kappa $ sentences of the second group are included, so the set of $x$ for which the corresponding ${c}_{x}$ must all appear in $T(\alpha )$ for some $$. But since $T$ has branches of arbitrary height, $T(\alpha )\vDash \mathrm{\Sigma}$.
Since $\kappa $ is weakly compact, it follows that $\mathrm{\Delta}$ also has a model, and that model obviously has a set of ${c}_{x}$ such that $B({c}_{x})$ whose corresponding elements of $T$ intersect every level and are compatible, therefore forming a cofinal branch of $T$, proving that $T$ is not Aronszajn.
Title  weakly compact cardinals and the tree property 

Canonical name  WeaklyCompactCardinalsAndTheTreeProperty 
Date of creation  20130322 12:52:51 
Last modified on  20130322 12:52:51 
Owner  Henry (455) 
Last modified by  Henry (455) 
Numerical id  7 
Author  Henry (455) 
Entry type  Result 
Classification  msc 03E10 
Related topic  TreeProperty 
Related topic  Aronszajn 