$\left|\Omega\right|$ positions ${i}$ of $W$ into equivalence classes $\AC_\alpha$ as follows: $i\AC_\alpha j$ if $B^\alpha(i)=B^\alpha(j).$

For any positive integers $a<b$ we have $\psi(a) \leqslant \psi(b).$

\[c:lem2\] For any positive integers $a, k$ we have $$\psi(a)\leqslant p_{n,d}^k\psi(k\cdot a)+k\cdot a$$

Consider the least representative in each class of $\AC_{k\cdot a}$. We get a sequence of positions $\{i_j\}.$ Now consider all $i_j$ and $B^{k\cdot a}(i_j)$ from the same equivalence class of $\AC_a.$ Suppose it consists of $B^{k\cdot a}(i_j)$ for $i_j\in[b, c).$ Let $\{i_j\}'$ denote the segment of the sequence $\{i_j\},$ such that $i_j\in[b, c-k\cdot a).$

Fix a positive integer $r, 1\leqslant r\leqslant p_{n,d}.$ All $k\cdot a$-beginnings of color $r$ that begin from positions of the word $W$ in $\{i_j\}'$ will be called representatives of type $r$. All representatives of type $r$ are pairwise distinct because they begin from the least positions in equivalence classes of $\AC_{k\cdot a}.$ Divide each representative of type $r$ into $k$ segments of length $a$. Enumerate segments inside each representative of type $r$ from left to right by integers from zero to $(k-1)$. If there exist $(p_{n,d}+1)$ representatives of type $r$ with the same first $(t-1)$ segments but with pairwise different $t$th segments where $1\leqslant t\leqslant k-1$ then there 