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Is it possibile to find a function $f:\mathbb N^2\to\mathbb N$

Source: KoMaL A. 821

April 12, 2022
functionkomalalgebra

Problem Statement

a) Is it possible to find a function f:N2→Nf:\mathbb N^2\to\mathbb N such that for every function g:N→Ng:\mathbb N\to\mathbb N and positive integer MM there exists n∈Nn\in\mathbb N such that set {k∈N:f(n,k)=g(k)}\left\{k\in \mathbb N : f(n,k)=g(k)\right\} has at least MM elements? b) Is it possible to find a function f:N2→Nf:\mathbb N^2\to\mathbb N such that for every function g:N→Ng:\mathbb N\to\mathbb N there exists n∈Nn\in \mathbb N such that set {k∈N:f(n,k)=g(k)}\left\{k\in\mathbb N : f(n,k)=g(k)\right\} has an infinite number of elements?
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