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$IP$ sets play fundamental role in arithmetic Ramsey theory. A set is calledan additive $IP$ set if it is of the form $FS\left(\langle x_{n}\rangle_{n\in\mathbb{N}}\right)=\left\{ \sum_{t\in H}x_{t}:H\right.$ is a nonempty finitesubset of $\left.\mathbb{N}\right\}$, whereas it is called a multiplicative$IP$ set if it is of the form $FP\left(\langle x_{n}\rangle_{n\in\mathbb{N}}\right)=\left\{ \prod_{t\in H}x_{t}:H\right.$ is a nonempty finitesubset of $\left. \mathbb{N}\right\}$ for some injective sequence $\langlex_{n}\rangle_{n\in \mathbb{N}}.$ An additive $IP^{\star}$ (resp. multiplicative$IP^{\star}$) set is a set which intersects every additive $IP$ set (resp.multiplicative $IP$ set). In \cite{key-1}, V. Bergelson and N. Hindman studiedhow rich additive $IP^{\star}$ sets are. They proved additive $IP^{\star}$ sets($AIP^{\star}$ in short) contain finite sums and finite products of a singlesequence. An analogous study was made by A. Sisto in\cite{key-3}, where heproved that multiplicative $IP^{\star}$ sets ($MIP^{\star}$ in short) containexponential tower\footnote{will be defined later} and finite product of asingle sequence. However exponential patterns can be defined in two differentways. In this article we will prove that $MIP^{\star}$ sets contain twodifferent exponential patterns and finite product of a single sequence. Thisimmediately improves the result of A. Sisto. We also construct a $MIP^\star$set, not arising from the recurrence of measurable dynamical systems.Throughout our work we will use the machinery of the algebra of theStone-\v{C}ech Compactification of $\mathbb{N}$.

Combined exponential patterns in multiplicative $IP^{\star}$ sets