A combinatorial approach to exponential patterns in multiplicative $IP^{\star}$ sets in $\mathbb{N}$

In [On $IP^{\star}$sets and central sets, Combinatorica, 14 (1994) 269-277], N. Hindman and V.Bergelson proved additive $IP^{\star}$-sets contain finite sums and finite products of a single sequence. An analogous study was made by A. Sisto in [Exponential triples, Electronics Journal of Combinatorics, 18 (2011), no. 147], where he proved that multiplicative $IP^{\star}$-sets contain exponential $IP$ of type $I$ and finite sums of a single sequence as well as exponential $IP$ of type $II$ and finite products of another single sequence, using the algebra in the Stone-\v{C}ech Compactification of discrete semigroups. In this article, we will provide a combinatorial proof of the result of A. Sisto.