$IP$ rich ultrafilters and new partition regular equations

In $1979$, using the theory of ultrafilters, N. Hindman proved that for every finite coloring of N, there exists a color that contains both additive and multiplicative $IP$ sets. Later, in $1993$, V. Bergelson and N. Hindman found an elementary proof. In this article, we prove that the partial product of these two $IP$ sets also lies in the same color. As an immediate consequence, we have the equation a+b=c· d partition regular. This was a conjecture of P. Csikv\'{a}ri, K. Gyarmati, and A. S\'{a}rk\"{o}zy, which was solved by V. Bergelson and N. Hindman independently. In this article, we separately give a very short proof of this conjecture. Then we extend these results to the flavor of Rado systems.