Tolerance in Helly-Type Theorems

In this paper we introduce the notion of tolerance in connection with Helly-type theorems and prove, using the Erdős–Gallai theorem, that any Helly-type theorem can be generalized by relaxing the assumptions and conclusion, allowing a bounded number of exceptional sets or points. In particular, we analyze some of the classical Helly-type theorems, such as Caratheodory’s and Tverberg’s theorems, as well as some other interesting ones.