2024/11/21 by Sayok Chakravarty, Chakravarty, Sayok, Dhruv Mubayi +1
Physics and Astronomy · #05B05 #05B25 #05D05 #Combinatorics (math.CO) #FOS: Mathematics #Nonlinear Waves and Solitons
paper · pdf · doi:10.48550/arxiv.2411.14634
openalex publication_date 2024/11/21 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Fix an integer s ≥ 2. Let P be a set of n points and let L be a set of lines in a linear space such that no line in L contains more than (n-1)/(s-1) points of P. Suppose that for every s-set S in P, there is a pair of points in S that lies in a line from L. We prove that |L| ≥ (n-1)/(s-1)+s-1 for n large, and this is sharp when n-1 is a multiple of s-1. This generalizes the de Bruijn-Erd\H os theorem which is the case s=2. Our result is proved in the more general setting of linear hypergraphs.