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SCHUR’S COLOURING THEOREM FOR NONCOMMUTING PAIRS

Part of: Graph theory

Published online by Cambridge University Press:  11 April 2019

TOM SANDERS Open the ORCID record for TOM SANDERS [Opens in a new window]
TOM SANDERS*
Affiliation:
Mathematical Institute, University of Oxford, Radcliffe Observatory Quarter, Woodstock Road, Oxford OX2 6GG, UK email [email protected]
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Abstract

For $G$ a finite non-Abelian group we write $c(G)$ for the probability that two randomly chosen elements commute and $k(G)$ for the largest integer such that any $k(G)$-colouring of $G$ is guaranteed to contain a monochromatic quadruple $(x,y,xy,yx)$ with $xy\neq yx$. We show that $c(G)\rightarrow 0$ if and only if $k(G)\rightarrow \infty$.

Keywords

Schur’s colouring theoremcommuting probabilitynon-Abelian group

MSC classification

Secondary: 05C15: Coloring of graphs and hypergraphs
Type
Research Article
Information
Bulletin of the Australian Mathematical Society , Volume 100 , Issue 3 , December 2019 , pp. 446 - 452
Copyright
© 2019 Australian Mathematical Publishing Association Inc. 

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References

Ajtai, M. and Szemerédi, E., ‘Sets of lattice points that form no squares’, Studia Sci. Math. Hungar. 9 (1974), 9–11.Google Scholar
Austin, T., ‘Ajtai–Szemerédi theorems over quasirandom groups’, in: Recent Trends in Combinatorics, IMA Volumes in Mathematics and its Applications, 159 (Springer, Cham, 2016), 453–484.Google Scholar
Bergelson, V. and McCutcheon, R., ‘Recurrence for semigroup actions and a non-commutative Schur theorem’, in: Topological Dynamics and Applications (Minneapolis, MN, 1995), Contemporary Mathematics, 215 (American Mathematical Society, Providence, RI, 1998), 205–222.Google Scholar
Bergelson, V., McCutcheon, R. and Zhang, Q., ‘A Roth theorem for amenable groups’, Amer. J. Math. 119(6) (1997), 1173–1211.Google Scholar
Bergelson, V. and Tao, T. C., ‘Multiple recurrence in quasirandom groups’, Geom. Funct. Anal. 24(1) (2014), 1–48.Google Scholar
Eberhard, S., ‘Commuting probabilities of finite groups’, Bull. Lond. Math. Soc. 47(5) (2015), 796–808.Google Scholar
Fox, J., ‘A new proof of the graph removal lemma’, Ann. of Math. (2) 174(1) (2011), 561–579.Google Scholar
Gowers, W. T., ‘Quasirandom groups’, Combin. Probab. Comput. 17(3) (2008), 363–387.Google Scholar
Gustafson, W. H., ‘What is the probability that two group elements commute?’, Amer. Math. Monthly 80(9) (1973), 1031–1034.Google Scholar
Hegarty, P., ‘Limit points in the range of the commuting probability function on finite groups’, J. Group Theory 16(2) (2013), 235–247.Google Scholar
Kemperman, J. H. B., ‘On complexes in a semigroup’, Nederl. Akad. Wetensch. Proc. Ser. A. 59 (1956), 247–254.Google Scholar
McCutcheon, R., ‘Non-commutative Schur configurations in finite groups’, Preprint, http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.538.5511&rep=rep1&type=pdf.Google Scholar
McCutcheon, R., ‘Monochromatic permutation quadruples—a Schur thing in S n ’, Amer. Math. Monthly 119(4) (2012), 342–343.Google Scholar
Neuman, P. M., ‘Two combinatorial problems in group theory’, Bull. Lond. Math. Soc. 21(5) (1989), 456–458.Google Scholar
Olson, J. E., ‘Sums of sets of group elements’, Acta Arith. 28(2) (1975/76), 147–156.Google Scholar
Olson, J. E., ‘On the sum of two sets in a group’, J. Number Theory 18(1) (1984), 110–120.Google Scholar
Sanders, T., ‘Solving xz = y 2 in certain subsets of finite groups’, Q. J. Math. 68(1) (2017), 243–273.Google Scholar
Schur, I., ‘Über die Kongruenz x m + y m ≡ z m (mod.p)’, Jahresber. Dtsch. Math.-Ver. 25 (1916), 114–117.Google Scholar
Shkredov, I. D., ‘On a generalization of Szemerédi’s theorem’, Proc. Lond. Math. Soc. (3) 93(3) (2006), 723–760.Google Scholar
Shkredov, I. D., ‘On a problem of Gowers’, Izv. Ross. Akad. Nauk Ser. Mat. 70(2) (2006), 179–221.Google Scholar
Shkredov, I. D., ‘On a two-dimensional analogue of Szemerédi’s theorem in abelian groups’, Izv. Ross. Akad. Nauk Ser. Mat. 73(5) (2009), 181–224.Google Scholar
Solymosi, J., ‘Roth-type theorems in finite groups’, European J. Combin. 34(8) (2013), 1454–1458.Google Scholar
Tao, T. C., ‘A variant of the hypergraph removal lemma’, J. Combin. Theory Ser. A 113(7) (2006), 1257–1280.Google Scholar