  In this paper we provide two results. The first one consists an infinitary
version of the Furstenberg-Weiss Theorem. More precisely we show that every
subset $A$ of a homogeneous tree $T$ such that $\frac{|A\cap
T(n)|}{|T(n)|}\geq\delta$, where T(n) denotes the $n$-th level of $T$, for all
$n$ in a van der Waerden set, for some positive real $\delta$, contains a
strong subtree having a level sets which forms a van der Waerden set.
  The second result is the following. For every sequence $(m_q)_{q}$ of
positive integers and for every real $0<\delta\\leq1$, there exists a sequence
$(n_q)_{q}$ of positive integers such that for every $D\subseteq
\bigcup_k\prod_{q=0}^{k-1}[n_q]$ satisfying $$\frac{\big{|}D\cap
\prod_{q=0}^{k-1} [n_q]\big{|}}{\prod_{q=0}^{k-1}n_q}\geq\delta$$ for every $k$
in a van der Waerden set, there is a sequence $(J_q)_{q}$, where $J_q$ is an
arithmetic progression of length $m_q$ contained in $[n_q]$ for all $q$, such
that $\prod_{q=0}^{k-1}J_q\subseteq D$ for every $k$ in a van der Waerden set.
Moreover, working in an abstract setting, we obtain $J_q$ to be any
configuration of natural numbers that can be found in an arbitrary set of
positive density.
