Exponential Schur and Hindman Theorem in Ramsey Theory

Abstract: Answering a conjecture of A. Sisto, J. Sahasrabudhe proved the exponential version of the Schur theorem: for every finite coloring of the naturals, there exists a monochromatic copy of ${x,y,x^y:x\neq y},$ which initiates the study of exponential Ramsey theory in arithmetic combinatorics. In this article, We first give two short proofs of the exponential Schur theorem, one using Zorn's lemma and another using $IP_r^\star$ van der Waerden's theorem. Then using the polynomial van der Waerden theorem iteratively we give a proof of the exponential Hindman theorem. Then applying our results we prove for every natural number $m,n$ the equation $x_n^{x_{n-1}^{\cdot^{\cdot^{\cdot^{x_1}}}}}=y_1\cdots y_m$ is partition regular, which can be considered as the exponential version of a more general version of the P. Csikv\'{a}ri, K. Gyarmati, and A. S\'{a}rk\"{o}zy conjecture, which was solved by V. Bergelson and N. Hindman independently. As a consequence of our results, we also prove that for every finite partition of $\mathbb{N},$ there exists two different sequences $\langle x_n\rangle_n$ and $\langle y_n\rangle_n$ such that both the multiplicative and exponential version of Hindman theorem generated by these sequences resp. are monochromatic, whereas in the counterpart in the finitary case, J. Sahasrabudhe proved that both sequences are the same. Our result can be considered as an exponential analog to the result of V. Bergelson and N. Hindman. We also prove that a large class of ultrafilters with certain properties do not exist, which could give us direct proof of the exponential Schur theorem. This result can be thought of as partial evidence of the nonexistence of Galvin-Glazer's proof of the exponential Hindman theorem.