Are $m$-transitive relations also $n$-transitive for $n \geq m$?

Question

Let $R$ be a binary relation on a set $A$, let $n$ be a positive integer, and let $x_1,...,x_{n+2}$ be a list of $n+2$ distinct variables. I define $R$ to be $n$-transitive if for all $x_1,...,x_{n+2}$ in $A$, $(R(x_1,x_2) \land ... \land R(x_{n+1},x_{n+2})) \rightarrow R(x_1,x_{n+2})$. So for example, a $1$-transitive relation is just an ordinary transitive relation, and a $2$-transitive relation is one that satisfies the formula $(R(x,y) \land R(y,z) \land R(z,w)) \rightarrow R(x,w)$ for all $x,y,z,w$. I have noticed that every $1$-transitive relation is $n$-transitive for all $n \geq 1$. So, then, is every $m$-transitive relations also $n$-transitive for all $n \geq m$? Or, is it only the case that, if the positive integer $m$ divides the positive integer $n$, then $m$-transitive relations are also $n$-transitive? If neither of the previous statements hold, can someone give me useful necessary and sufficient conditions on the set of ordered pairs of positive integers $(m,n)$ such that every $m$-transitive relation is an $n$-transitive relation?

Answer 1

Here's an example showing that $m\mid n$ is necessary: let $R_m$ be the relation on the integers defined by $$ R_m(x,y) \iff x-y\equiv1\pmod m. $$ Then any set of $n+2$ consecutive integers is a counterexample to $R_m$ being $n$-transitive if $m\nmid n$.