A partition problem on colored sets

Let f1 and f2 be functions from a pair of finite sets X1 and X2, respectively, to another finite set Y. We say that f1 and f2 are isomorphic if there exists a permutation σ of Y such that |f1−1(y)|=|f2−1(σ(y))| for any y∈Y. For a function f defined on a finite set X, let r(f) be the smallest nonnegative integer r such that there exists a partition X=X0∪X1∪X2 satisfying (1) f|X1 and f|X2 are isomorphic and (2) |X0|=r. Let k⩾1. We define N(k) as the smallest integer n such that there exists a function f defined on an n-set satisfying r(f)⩾k. It is shown that N(k)=∑i=1kci, where c1=1, c2=3 and ci=ci−1+ci−2+3 for i⩾3.