Let $\mathcal G=(G_n)_n$ be a family of expanders; i.e. each $G_n=(V_n,E_n)$ is a finite connected d-regular graph of order say $\alpha(n)$, with $\alpha(n)\to\infty$, and each isoperimetric constant $\iota_n$ is bounded below by some positive real number $\epsilon$.
Let me suppose that $\alpha(n)=2n$, just to keep notation simpler. Let $A\subseteq V_n$ containing $n+1$ vertices. Denote by $M(A)$ the maximal distance of a vertex of $V_n$ from $A$.
Question: Can we found a uniform bound $M(A)\leq K$?
Here uniform means that $K$ may depend on the regularity $d$, on $\epsilon$, on other stuff, but neither on $A$ nor on $n$.
My intuition, maybe wrong, is that expanders have strong connectivity properties and therefore, taking $n+1$ vertex, the remaining $n-1$ cannot stay too far away.
Thank you in advance,