The authors give a partial answer to the following problem: Let \(X\) be a compact space with the fixed point property and \({\mathcal F}\) a family of continuous mappings from \(X\) to \(X\) such that \(|{\mathcal F}|\geq{\mathfrak c}\). In which conditions on \({\mathcal F}\) and \(X\) there is a subfamily \({\mathcal F}_0\subset{\mathcal F}\) of cardinality \(\geq{\mathfrak c}\) with a common fixed point?