The following question is inspired by [Function with vector space](https://mathoverflow.net/questions/362400/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.



On a property possibly separating countable and not countable cardinals
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](https://mathoverflow.net/users/2689/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 $f(x,y)\gt g(x)+g(y)$.


