Additive and multiplicative structures in completely I-large* sets

It is known that for any IP* set A and any sequence (xn)infinity n=1 in (N, +), there exists a sum subsystem of (xn)infinity n=1 whose finite sums and finite products are in A. Similar results have been established for some other combinatorial notions. In this article, we introduce a general notion called completely I-large* sets, which can be IP* sets, central* sets or C*-sets when I is the corresponding ideal. And we establish a result for this notion to unify all known results. Furthermore, we apply this general result to another notion - quasi-central* sets to obtain corresponding combinatorial results. (c) 2024 Elsevier B.V. All rights reserved.