The Rado Multiplicity Problem in Vector Spaces over Finite Fields

2023/04/01 by Rué, Juanjo, Spiegel, Christoph · 1 citation
#05D10 #11B25 #11B75 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT) #Optimization and Control (math.OC)

paper · doi:10.48550/arxiv.2304.00400

Abstract

We study an analogue of the Ramsey multiplicity problem for additive structures, in particular establishing the minimum number of monochromatic 3-APs in 3-colorings of \mathbbF₃ⁿ as well as obtaining the first non-trivial lower bound for the minimum number of monochromatic 4-APs in 2-colorings of \mathbbF₅ⁿ. The former parallels results by Cumings et al (2013) in extremal graph theory and the latter improves upon results of Saad and Wolf (2017) The lower bounds are notably obtained by extending the flag algebra calculus of Razborov (2007) to additive structures in vector spaces over finite fields.

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• Lower and Upper Bounds for Small Canonical and Ordered Ramsey Numbers

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