For two set-valued mappings \(\Phi,\Psi: X\multimap Y\) the coincidence set is defined as \(\text{Coin}(\Phi,\Psi):=\{\xi\in X \mid \Phi(\xi)\cap\Psi(\xi)\not=\emptyset\}\). Let \(\alpha>0\) and \(\beta\geq0\). A map \(\Psi:X\multimap Y\) between metric spaces is called an \(\alpha\)-covering if \(B_Y(\Psi(x_0);\alpha r)\subset \Psi(B_X(x_0;r))\) for all \(x_0\in X\) and \(r>0\) where of course, \(B_X\) and \(B_Y\) denote the corresponding closed balls. \(\Psi\) is called \(\beta\)-Lipschitz if \(h_Y(\Phi(x),\Phi(u))\leq\beta\rho_X(x,u)\) for all \(x,u\in X\) where \(h_Y\) denotes the Hausdorff-distance in \(Y\) and \(\rho_X\) the metric in \(X\). Let now be \(\Omega\) be an arbitrary set (finite or infinite) and let \(\Phi:X\multimap Y\) be a closed \(\alpha\)-covering and let \(\Psi:X\multimap Y\) be closed and \(\beta\)-Lipschitz. Assume that \(\alpha>\beta\) and that at least one of \(\Phi\) and \(\Psi\) has a graph which is a complete metric space. The authors use the following rather technical assumption: There is an injective mapping \(\Omega\to X\), \(j\mapsto x_j\), and for each \(j\in\Omega\) there is an \(\epsilon_j>0\) such that for \(i\not=j\) we have that \(\rho(x_i,x_j)\geq\frac{\text{dist}_Y(\Psi(x_i),\Phi(x_i))+\text{dist}_Y(\Psi(x_j),\Phi(x_j))}{\alpha-\beta}+\epsilon_i+\epsilon_j\). Let then \((y_j)_{j\in\Omega}\) be a family in \(Y\) such that \(y_j\in\Psi(x_j)\). Then there exist families \((\xi_j)_{j\in\Omega}\) in \(X\) and \((\eta_j)_{j\in\Omega}\) in \(Y\) such that \((\xi_j,\eta_j)\) belongs to the intersection of the graphs of \(\Phi\) and \(\Psi\), \(\xi_i\not=\xi_j\) for \(i\not=j\), and the cardinality of \(\text{Coinc}(\Psi,\Phi)\) is not smaller than the cardinality of \(\Omega\). There are several results in the same spirit where special consideration is given to the case of normed spaces.