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A partition property of simplices in Euclidean space
Author(s):
P.
Frankl;
V.
Rödl
Journal:
J. Amer. Math. Soc.
3
(1990),
1-7.
MSC:
Primary 52A37;
Secondary 05A99
MathSciNet review:
1020148
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Abstract:
Given the vertex set  of a nondegenerate simplex in  , it is shown that for some positive  and every partition of  into fewer than  parts, one of the parts must contain a set congruent to  . This solves a fifteen-year-old problem of Erdös et al. [E].
References:
-
- [E]
- P. Erdös, R. L. Graham, P. Montgomery, B. L. Rothschild, J. Spencer, and E. Straus, Euclidean Ramsey theorems. I, J. Combin. Theory Ser. A 14 (1973), 341-363; Euclidean Ramsey theorems. II, III, Infinite and Finite Sets (A. Hajnal, R. Rado, and V. T. Sós, eds.), North-Holland, Amsterdam, 1975, pp. 529-557 and 559-583. MR 0316277 (47:4825)
- [FR1]
- P. Frankl and V. Rödl, All triangles are Ramsey, Trans. Amer. Math. Soc. 297 (1986), 777-779. MR 854099 (88d:05018)
- [FR2]
- -, Forbidden intersections, Trans. Amer. Math. Soc. 300 (1987), 259-286. MR 871675 (88m:05003)
- [FRo]
- P. Frankl and I. G. Rosenberg, A finite set intersection theorem, European J. Combin. 2 (1981), 127-129. MR 622077 (82j:05002)
- [FW]
- P. Frankl and R. M. Wilson, Intersection theorems with geometric consequences, Combinatorica 1 (1981), 357-368. MR 647986 (84g:05085)
- [G1]
- R. L. Graham, Old and new Euclidean Ramsey theorems, discrete geometry and convexity, Ann. New York Acad. Sci. 440 (1985), 20-30. MR 809188 (87b:05021)
- [G2]
- -, Rudiments of Ramsey theory, CBMS Regional Conf. Ser. in Math., no. 45, Amer. Math. Soc., Providence, RI, 1981. MR 608630 (82j:05018)
- [R]
- V. Rödl, On a problem in combinatorial geometry, Discrete Math. 45 (1983), 129-131. MR 700858 (84m:05029)
- [S]
- I. J. Schönberg, Metric spaces and positive definite functions, Trans. Amer. Math. Soc. 44 (1938), 522-536. MR 1501980
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Additional Information:
DOI:
10.1090/S0894-0347-1990-1020148-2
PII:
S0894-0347-1990-1020148-2
Copyright of article:
Copyright
1990,
American Mathematical Society
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