Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

 
 

 

Avoidable algebraic subsets
of Euclidean space


Author: James H. Schmerl
Journal: Trans. Amer. Math. Soc. 352 (2000), 2479-2489
MSC (1991): Primary 03E15, 04A20
DOI: https://doi.org/10.1090/S0002-9947-99-02331-4
Published electronically: July 9, 1999
MathSciNet review: 1608502
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Fix an integer $n\ge 1$ and consider real $n$-dimensional $\mathbb{R}^n$. A partition of $\mathbb{R}^n$ avoids the polynomial $p(x_0,x_1,\dotsc,x_{k-1})\in\mathbb R[x_0,x_1,\dotsc,x_{k-1}]$, where each $x_i$ is an $n$-tuple of variables, if there is no set of the partition which contains distinct $a_0,a_1,\dotsc,a_{k-1}$ such that $p(a_0,a_1,\dotsc,a_{k-1})=0$. The polynomial is avoidable if some countable partition avoids it. The avoidable polynomials are studied here. The polynomial $\|x-y\|^2-\|y-z\|^2$ is an especially interesting example of an avoidable one. We find (1) a countable partition which avoids every avoidable polynomial over $Q$, and (2) a characterization of the avoidable polynomials. An important feature is that both the ``master'' partition in (1) and the characterization in (2) depend on the cardinality of $\mathbb R$.


References [Enhancements On Off] (What's this?)

  • 1. R. Benedetti and J.-J. Risler, Real Algebraic and Semi-algebraic Sets, Hermann, Paris, 1990. MR 91j:14045
  • 2. J. Ceder, Finite subsets and countable decompositions of Euclidean spaces, Rev. Roumaine Math. Pures Appl. 14 (1969), 1247-1251. MR 41:1958
  • 3. R. O. Davies, Partitioning the plane into denumerably many sets without repeated distances, Proc. Cambridge Philos. Soc. 72 (1972), 179-183. MR 45:3662
  • 4. P. Erd\H{o}s and A. Hajnal, On chromatic number of graphs and set systems, Acta Math. Hungar. 17 (1966), 61-99. MR 41:8294
  • 5. P. Erd\H{o}s, A. Hajnal, A. Máté and R. Rado, Combinatorial Set Theory; Partition Relation for Cardinals, North-Holland, Amsterdam, 1984. MR 87g:04002
  • 6. P. Erd\H{o}s and S. Kakutani, On non-denumerable graphs, Bull. Amer. Math. Soc. 49 (1943), 457-461. MR 4:249f
  • 7. P. Erd\H{o}s and P. Komjáth, Countable decompositions of $\mathbb R^2$ and $\mathbb R^3$, Discrete & Comput. Geom. 5 (1990), 325-331. MR 91b:04002
  • 8. P. Komjáth, Tetrahedron free decompositions of $\mathbb R^3$, Bull. London Math. Soc. 23 (1991), 116-120. MR 92i:52008
  • 9. P. Komjáth, The master coloring, Comptes Rendus Mathématique de l'Academie des Sciences, la Société Royale du Canada 14 (1992), 181-182. MR 94a:05076
  • 10. P. Komjáth, A decomposition theorem for $\mathbb R^n$, Proc. Amer. Math. Soc. 120 (1994), 921-927. MR 94h:04005
  • 11. P. Komjáth, Partitions of vector spaces, Period. Math. Hungar. 28 (1994), 187-193. MR 95k:04004
  • 12. P. Komjáth, personal letter to the author, 1992.
  • 13. K. Kunen, Partitioning Euclidean space, Math. Proc. Cambridge Philos. Soc. 102 (1987), 379-383. MR 88i:05014
  • 14. J. H. Schmerl, Partitioning Euclidean space, Discrete & Comp. Geom. 10 (1993), 101-106. MR 94e:05262
  • 15. J. H. Schmerl, Triangle-free partitions of Euclidean space, Bull. London Math. Soc. 26 (1994), 483-486. MR 95i:52019
  • 16. J. H. Schmerl, Countable partitions of Euclidean space, Math. Proc. Cambridge Philos. Soc. 120 (1996), 7-12. MR 96k:52011
  • 17. L. van den Dries, Alfred Tarski's elimination theory for real closed fields, J. Symbolic Logic 53 (1988), 7-19. MR 89h:01040

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (1991): 03E15, 04A20

Retrieve articles in all journals with MSC (1991): 03E15, 04A20


Additional Information

James H. Schmerl
Affiliation: Department of Mathematics, University of Connecticut, Storrs, Connecticut 06269-3009
Email: schmerl@math.uconn.edu

DOI: https://doi.org/10.1090/S0002-9947-99-02331-4
Keywords: Algebraic sets, avoidable polynomials, infinite combinatorics
Received by editor(s): November 5, 1997
Published electronically: July 9, 1999
Article copyright: © Copyright 2000 American Mathematical Society

American Mathematical Society