Let $\kappa$ be weakly compact, and let $(T, <_T)$ be a tree of height $\kappa$ such that each level of $T$ has size $< \kappa$. Assume $T = \kappa$. We extend the partial ordering $<_T$ of $\kappa$ as follows - if $\alpha <_T \beta$ then say $\alpha \prec \beta$. If $\alpha, \beta$ are incomparable then let $\xi$ be the first level where the predecessors of $\alpha, \beta$ differ, and if $\alpha_{\xi} < \beta_{\xi}$ in the usual ordering of $\kappa$, say $\alpha \prec \beta$.
Define $F: [\kappa]^2 \to \{ 0,1\}$ by $F(\{ \alpha, \beta\}) = 1$ if and only if $<$ and $\prec$ agree on $\alpha, \beta$. Assume this is onto and by weak compactness there is a homogeneous set $H \subseteq \kappa$ of size $\kappa$.
Define a set $B$ to be the collection of all $x \in \kappa$ such that $\{ \alpha \in H \mid x <_T \alpha\}$ has size $\kappa$. My question is, why is there an element of $B$ on every level of $T$? It has something to do with the fact that each level of $T$ has size $< \kappa$, but I can't see why. Any help would be appreciated, thank you.