say we have a sequence $x_1,x_2,x_3...$ with y terms, and one rule that for $i\ge1$ and $n\ge0$ $$x_i,x_{i+1}...x_{i+n}\ne x_{i+n+1},x_{i+n+2}...x_{i+2n+1}$$ so for some example, when $y=1$, would be of length one, since if $i=1$ and $n=0$ then that states that you cannot have the same symbol twice in a row. for $y=2$, the longest possible is 3. without loss of generality we can assume 1 is first, then since you cant directly repeat, 12 is the only option and similarly 121 is the only next option, but when you try to add a next term 1211 is invalid $i=3$ and $n=0$ and 1212 is invalid $i=1$ $n=1$.
I have not yet calculated the longest possible $y=3$ but if ramsey theory works, there should be a longest sequence for any y value. here is one valid sequence for $y=3$ is 1213123132123. I am also intriged on how many possible "words" there are (for $y=2$ there are 6 (1,2,12,21,121,212))