# Diameter for permutations of bounded support

Let $$S\subset \textrm{Sym}(n)$$ be a set of permutations each of which is of bounded support, that is, each $$\sigma\in S$$ moves $$O(1)$$ elements of $$\{1,2,\dotsc,n\}$$. Let $$\Gamma$$ be the graph whose set of vertices is the set of ordered pairs $$(i,j)$$, $$1\leq i,j\leq n$$, $$i\ne j$$, and whose edges are $$((i,j), (i,j)^\sigma)$$, $$\sigma\in S$$. (In other words, $$\Gamma$$ is a Schreier graph.) Assume $$\Gamma$$ is connected, i.e., $$S$$ generates a $$2$$-transitive group.

Must it be the case that $$\textrm{diam}(\Gamma) = O(n)$$? Could it be the case that $$\textrm{diam}(\Gamma)\gg n^{1+\delta}$$, $$\delta>0$$, or even $$\delta=1$$?

What are the answers to these questions if we assume that $$\langle S\rangle$$ is all of $$\textrm{Alt}(n)$$ or $$\textrm{Sym(n)}$$?