What question did this study set out to answer?

To generalize the de Bruijn–Erdős theorem by exploring conditions for points and lines in linear spaces.

February 8, 2026

Combining the Theorems of Turán and de Bruijn–Erdős

Key Points

To generalize the de Bruijn–Erdős theorem by exploring conditions for points and lines in linear spaces.
Fixed an integer for analysis of point-line relationships
Defined a set of points and lines in a linear space
Considered conditions for pairs of points lying on lines
Proved results are sharp when the integer is a multiple of a specified value
Demonstrated that conditions hold for large integers
Results applicable in the context of linear hypergraphs

Abstract

ABSTRACT 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.

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