The constant of point-line incidence constructions

2022/10/10 by Balko, Martin, Sheffer, Adam, Tang, Ruiwen
#Combinatorics (math.CO) #FOS: Mathematics

paper · doi:10.48550/arxiv.2210.04821

Abstract

We study a lower bound for the constant of the Szemerédi-Trotter theorem. In particular, we show that a recent infinite family of point-line configurations satisfies I(\mathcal P,\mathcal L)≥ (c+o(1)) |\mathcal P|^2/3|\mathcal L|^2/3, with c≈ 1.27. Our technique is based on studying a variety of properties of Euler's totient function. We also improve the current best constant for Elekes's construction from 1 to about 1.27. From an expository perspective, this is the first full analysis of the constant of Erd\H os's construction.

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