On a property possibly separating countable and not countable cardinals The following question is inspired by Function with vector space , which has been closed for unknown reason and which may have a wellknown answer. Is the following true?
Let $X$ be an uncountable set. Then there is a function $f \colon X \times X \to \mathbb{N}$ such that for any function $g \colon X \to \mathbb{N}$ there is $(x,y) \in X^2$ with $f(x,y) > g(x) + g(y)$.
It is easy to show that this is false if $X$ is countable.
 A: Yes, such a function $f:X\times X\to\mathbb N$ exists if $X$ is uncountable. It will suffice to prove it for $X=\omega_1$. The following proof is based on the same idea as a comment by Ashutosh but uses elementary set theory instead of the result of Todorcevic.
For each ordinal $\alpha\in\omega_1$ choose an injective map $\psi_\alpha:\alpha\to\mathbb N$. Define a function $f:\omega_1\times\omega_1\to\mathbb N$ so that $f(\alpha,\beta)=\psi_\alpha(\beta)$ when $\beta\lt\alpha$. Now consider any function $g:\omega_1\to\mathbb N$. Then $g$ is bounded on some uncountable set $Y\subseteq\omega_1$; say $g(\xi)\le n\in\mathbb N$ for all $\xi\in Y$. Let $Z\subset Y$ be a set of order type $\omega+1$ and let $\alpha=\max Z$. Since $\{f(\alpha,\beta):\beta\in Z\cap\alpha\}=\{\psi_\alpha(\beta):\beta\in Z\cap\alpha\}$ is an infinite subset of $\mathbb N$, we can choose $\beta\in Z\cap\alpha$ with $f(\alpha,\beta)\gt2n\ge g(\alpha)+g(\beta)$.
More generally, if $|X|\gt\aleph_\alpha$, then there is a function $f:X\times X\to\omega_\alpha$ such that, for any function $g:X\to\omega_\alpha$, there are $x,y\in X$ with $f(x,y)\gt g(x)+g(y)$.
