Combining the Theorems of Turán and de Bruijn–Erdős

Fix an integer . Let be a set of points and let be a set of lines in a linear space such that no line in contains more than points of . Suppose that for every ‐set in , there is a pair of points in that lies in a line from . We prove that for large, and this is sharp when is a multiple of . This generalizes the de Bruijn–Erdős theorem, which is the case . Our result is proved in the more general setting of linear hypergraphs.