Bell, Jason; Funk, Daryl; Du Kim, Byoung; Mayhew, Dillon Effective versions of two theorems of Rado. (English) Zbl 1448.52011 Q. J. Math. 71, No. 2, 599-618 (2020). In 1957, Richard Rado proved that each representative matroid is representable over a finite field \(\mathbb{F}\) (see [R. Rado, Proc. Lond. Math. Soc. (3) 7, 300–320 (1957; Zbl 0083.02302)]). In the present article, the authors ask: if the matroid has \(n\) elements, then how large must \(\mathbb{F}\) be?Letting \(\mathcal{M}_n\) denote the family of representable matroids on \(n\) elements, define \begin{align*} c(n) &= \max\{c(M) : M\in\mathcal{M}_n\}\\ f(n) &= \max\{f(M) : M\in\mathcal{M}_n\}\,, \end{align*} where \(c(M)\) (resp., \(f(M)\)) denotes the smallest positive characteristic (resp., the smallest order) of a field over which \(M\) is representable.The authors prove interesting upper and lower bounds for \(c(n)\) and \(f(n)\). Similarly, they prove that each matroid \(M\) on \(n\) elements which is representable over a field of characteristic 0 must also be representable over \(\text{GF}(p)\) whenever \(p\) is a prime satisfying \[ \log_2\log_2\log_2 p > n^5\,. \] Reviewer: Thomas Britz (Sydney) MSC: 52B40 Matroids in convex geometry (realizations in the context of convex polytopes, convexity in combinatorial structures, etc.) 05B35 Combinatorial aspects of matroids and geometric lattices Keywords:representable matroid; extension field; finite field; bound Citations:Zbl 0083.02302 PDFBibTeX XMLCite \textit{J. Bell} et al., Q. J. Math. 71, No. 2, 599--618 (2020; Zbl 1448.52011) Full Text: DOI arXiv