This question is related to this one but feels more Ramsey-type, so perhaps it is easier. Let $S$ be a finite set, $|S|=k$. Suppose we color all subsets of $S$ in $1000$ colors. What is the maximal (in terms of $k$) guaranteed length $l=l(k)$ of a monochromatic sequence of pairwise different subsets $A_1,A_2,..., A_l$ such that $|A_i\setminus A_{i+1}|+|A_{i+1}\setminus A_i|\le 2$ for every $i$? Clearly if $A$ is a subset of $S$ such that all 2-element subsets of $A$ are monochromatic, then $l(n)\ge |A|-1$ (there is a sequence of 2-element subsets of $A$ which satisfies the above property). So $l(k)$ is at least as big as the corresponding number from the Ramsey theory. Is it much bigger? The number 1000 is of course "any fixed number".

** Update 1 ** Fedor and Tony showed below that $l(k)\ge k/1000$. Thus only the first question remains: What is $l(k)$? Is it exponential in $k$, for example?

** Update 2 ** Although the question I asked makes sense (see Update 1), I realized that it is not the question I meant to ask. Here is the correct question. Same assumptions: $|S|=k$, 1000 colors. We consider monochromatic sequences of pairwise different subsets ${\mathcal A}=A_1,A_2,...,A_l$, where $|A_i\setminus A_{i+1}|+|A_{i+1}\setminus A_i|\le 2$. For each of these sequences we compute $\chi({\mathcal A})=|A_1\setminus A_l|+|A_l\setminus A_1|$. Now the question: what is the maximal guaranteed $\chi({\mathcal A})$ in terms of $k$, call it $\chi(k)$? By Ramsey, this number grows with $k$. Indeed if we color just $s$-element subsets, we will be able (if $k\gg s$) to find a subset of size $2s$ where all subsets of size $s$ are colored with the same color; then we can find a monochromatic sequence of subsets of size $s$ with the above property and $\chi=2s$ because the first and the last subsets in that sequence are disjoint. The question is what is the growth rate of $\chi(k)$. The question is motivated by Justin Moore's answer
here.