I had the following task ony my set theory exam
Suppose $ \kappa $ is an uncountable cardinal and $ \mathcal{F} $ is an uncountable set of partial finite functions from $ \kappa $ to $ \omega $, i.e. for each $ f \in \mathcal{F} $ its domain $D_f $is a finite subset of $ \kappa $ and $ f : D_f \to \omega = \mathbb{N}$ Prove that there exist $ f,g \in \mathcal{F} $ such that $$\forall_{x \in D_f \cap D_g} f(x) = f(y)$$
I thought the answer was an obvious implication of the $ \Delta$-lemma, but it seems that's not the case. Here's my (possibly outrageously naive) reasoning:
Treating functions as sets, $ \mathcal{F} $ is an uncountable collection of finite sets, which means (by the $ \Delta $-lemma) there exists an uncountable subcollection $ \mathcal{F}' \subset \mathcal{F} $ such that for all $ f\neq g, f,g \in \mathcal{F}'$ it holds that $ f \cap g = \Delta = \operatorname{const} $, which means that for any $ f,g \in \mathcal{F}' $ $$f \cap g = \{(x,y): x \in D_f \cap D_g \wedge y = f(x) = g(x)\}$$
And since $ \mathcal{F}' $ is uncountable, there are two such functions
I would appreciate some explanation
This is indeed a $\Delta$-system problem; however, the solution you've given does not work.
The problem is that saying that a set $F$ of finite partial functions forms a $\Delta$-system when we view each function as a set, does not mean that those functions are compatible! E.g. consider $f=\{(0, 0)\}$ and $g=\{(0, 1)\}$. Then $f\cap g=\emptyset$, but $f$ and $g$ are not compatible: there is an $x\in D_f\cap D_g$ (namely $x=0$) such that $f(x)\not=g(x)$.
Instead, we need to apply the $\Delta$-system lemma to the domains of the functions. We get an uncountable family $F$, and a finite set $D$, such that for $f, g\in F$ we have $dom(f)\cap dom(g)=D$. Now this does not guarantee compatibility; however, we can apply the pigeonhole principle - how many possibilities are there for $f\upharpoonright D$?