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다음과 같이 정의한 f(d)로 표현할 수 있다.
f(d)는 d차원에서 델타가 1 인 집합을 f(d) 개의 델타가 1 미만인 closed 집합으로 나누는 최소값이다.
이 guess는 보르스키에 의해提出되었으며, 2 차원 및 3 차원의 경우와 고른 조밀체, 연속체 등 특수한 경우를 제외하고는 알려진 바가 없었다.
본 논문에서 우리는 f(d)≥(1.2)^2 d를 증명하였다. 이를 증명하기 위해 우리의 방법은 2^4k개의 {0,1} 벡터를 m = 4k 개의 집합으로 나누고, 두 집합의 공통요소가 k개인 경우를 찾는 것이다.
이러한 방식으로 f(d)≥(1.2)^2 d를 증명하였다. 이 방법은 보르스키 guess에 대한 첫 번째 반례를 제공한다.
영어 요약 시작:
The paper presents a counterexample to Borsuk's conjecture. This conjecture was about partitioning every set of diameter one in Rd into d+1 closed sets of diameter smaller than one.
We define f(d) as the smallest number so that every set in Rd of diameter 1 can be partitioned into f(d) sets of diameter smaller than 1.
Borsuk's conjecture was introduced by Borsuk and it remained open for many years until we proved that f(d)≥(1.2)^2 d.
Our proof is based on finding a way to divide the set of all {0,1} vectors into m=4k sets such that any two sets have k elements in common.
This approach gives us the first counterexample to Borsuk's conjecture.
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arXiv:math/9307229v1 [math.MG] 1 Jul 1993APPEARED IN BULLETIN OF THEAMERICAN MATHEMATICAL SOCIETYVolume 29, Number 1, July 1993, Pages 60-62A COUNTEREXAMPLE TO BORSUK’S CONJECTUREJEFF KAHN AND GIL KALAIAbstract. Let f(d) be the smallest number so that every set in Rd of diam-eter 1 can be partitioned into f(d) sets of diameter smaller than 1.
Borsuk’sconjecture was that f(d)=d+1. We prove that f(d)≥(1.2)√d for large d.1.
IntroductionSixty years ago Borsuk [2] raised the following question.Problem 1 (Borsuk). Is it true that every set of diameter one in Rd can be par-titioned into d + 1 closed sets of diameter smaller than one?
The conjecture thatthis is true has come to be called Borsuk’s conjecture.Let f(d) be the smallest number so that every set in Rd of diameter 1 can bepartitioned into f(d) sets of diameter smaller than 1. The vertices of the regularsimplex in Rd show that f(d) ≥d + 1.
(Another example showing this is, bythe Borsuk-Ulam theorem, the d-dimensional Euclidean ball. )The assertion ofBorsuk’s conjecture was proved in dimensions 2 and 3 and in all dimensions forcentrally symmetric convex bodies and smooth convex bodies.
See [9, 1, 4] andreferences cited there. Lassak [14] proved that f(d) ≤2d−1 + 1, and Schramm[16] showed that for every ǫ, if d is sufficiently large, f(d) ≤(p(3/2) + ǫ)d. Adifferent proof of Schramm’s bound was given by Bourgain and Lindenstrauss [3].See [9, 1, 4] for surveys and many references on Borsuk’s problem.Borsuk’s conjecture seems to have been believed generally,and variousstronger conjectures have been proposed.
The possibility of a counterexample basedon combinatorial configurations was suggested by Erd˝os [6], Larman [15], and per-haps others. In 1965 Danzer [5] showed that the finite set K ⊆Rd consisting ofall {0, 1}-vectors of an appropriate weight cannot be covered by (1.003)d balls ofsmaller diameter.
Larman [13] observed that, for sets consisting of 0-1 vectors withconstant weight, Borsuk’s conjecture reduces to:Conjecture 1. Let K be a family of k-subsets of {1, 2, ..., n} such that every twomembers of K have t elements in common.
Then K can be partitioned into n partsso that in each part every two members have (t + 1) elements in common.Here we proveReceived by the editors June 30, 1992.1991 Mathematics Subject Classification. Primary 52A20; Secondary 05D05, 52C17.The first author was supported in part by BSF, NSF, and AFOSR.
The second author wassupported in part by GIF.c⃝1993 American Mathematical Society0273-0979/93 $1.00 + $.25 per page1
2JEFF KAHN AND GIL KALAITheorem 1. For large enough d, f(d) ≥(1.2)√d by constructing an appropriatefamily of sets.We need the following result of Frankl and Wilson [8].Theorem 2 (Frankl and Wilson).
Let k be a prime power and n = 4k. Let Kbe a family of n/2-subsets of {1, 2, ..., n}, so that no two sets in the family haveintersection of size n/4.
Then|K| ≤2 · n −1n/4 −1.This settled, in particular, a (much weaker) conjecture of Larman and Rogers[12] and implies that, if g(d) is the smallest number so that Rd can be colored byg(d) colors such that no two points of the same color are distance one apart, theng(d) ≥(1.2)d.Let us also mention the following related result conjectured by Erd˝os and provedby Frankl and R¨odl [7].Theorem 3 (Frankl and R¨odl). Let n be a positive integer divisible by four.
LetK be a family of n/2-subsets of {1, 2, ..., n} such that no two sets in the family haveintersection of size n/4. Then |K| ≤(1.99)n.2.
The constructionHowever contracted, that definition is the result of expanded meditation.—Herman Melville, Moby DickLet V = {1, 2, ..., m}, and m = 4k, and k is a prime power. Let W be the setof pairs of elements in V .
For every partition P = {A, B} of V let S(A, B) be thesets of all pairs which contain one element from A and one element from B. Let Kbe the family of all sets of pairs which correspond to partitions of V into two equalparts, i.e., K = {S(A, B) : |A| = 2k}.
Thus, K is a family of (m2/4)-subsets ofan m(m −1)/2-set. The smallest intersection between S(A, B) and S(C, D) occurswhen |A∩C| = k, and by the Frankl-Wilson theorem every subfamily of more than2 · m−1m/4−1sets in K contains two sets which realize the minimal distance.
Thus,K cannot be partitioned into fewer than12 mm/22 · m−1m/4−1parts so that the minimal intersection is not realized in any of the parts. Thisexpression is greater than (1.203)√d for sufficiently large d = (m2 )−1, and Theorem1 for general (large) d follows via the prime number theorem.
A COUNTEREXAMPLE TO BORSUK’S CONJECTURE33. Remarks1.
In view of Theorem 1, the upper bounds on f(d) cited earlier seem muchmore reasonable than formerly.It would be of considerable interest to have abetter understanding of the asymptotic behavior of log f(d). At the moment, wecannot distinguish the asymptotic behavior of f(d) from that of g(d).Also ofinterest would be counterexamples in small dimensions.
Our construction showsthat Borsuk’s conjecture is false for d = 1, 325 and for every d > 2, 014.2.Larman’s conjecture for t = 1 is open and still quite interesting, in partbecause of its similarity to the Erd˝os-Faber-Lov´asz conjecture.See [11, 10] forsome discussion and related results.3. Intersection properties of edge-sets of graphs were first studied by S´os; see[17] and references quoted therein.References1.
V. Boltjansky and I. Gohberg, Results and problems in combinatorial geometry, CambridgeUniv. Press, Cambridge, 1985.2.
K. Borsuk, Drei S¨atze ¨uber die n-dimensionale euklidische Sph¨are, Fund. Math.
20 (1933),177–190.3. J. Bourgain and J. Lindenstrauss, On covering a set in Rd by balls of the same diameter,Geometric Aspects of Functional Analysis (J. Lindenstrauss and V. Milman, eds.
), LectureNotes in Math., vol. 1469, Springer-Verlag, Berlin, 1991, pp.
138–144.4. H. Croft, K. Falconer, and R. Guy, Unsolved problems in geometry, Springer-Verlag, NewYork, 1991, pp.
123–125.5. L. Danzer, On the k-th diameter in Ed and a problem of Gr¨unbaum, Proc.
Colloq. onConvexity 1965 (W. Fenchel, ed.
), Københavns Univ. Math.
Inst., 1967.6. P. Erd˝os, My Scottish book “problems”, The Scottish Book, Mathematics from the ScottishCaf´e (R. D. Mauldin, ed.
), Birkh¨auser, 1981, pp. 35–43.7.
P. Frankl and V. R¨odl, Forbidden intersections, Trans. Amer.
Math. Soc.
300 (1987), 259–286.8. P. Frankl and R. Wilson, Intersection theorems with geometric consequences, Combinatorica1 (1981), 357–368.9.
B. Gr¨unbaum, Borsuk’s problem and related questions, Proc. Sympos.
Pure Math., vol. 7,Amer.
Math. Soc, Providence, RI, 1963.10.
J. Kahn and G. Kalai, A problem of F¨uredi and Seymour on covering intersecting familiesby pairs (to appear).11. J. Kahn and P. Seymour, A fractional version of the Erd˝os-Faber-Lov´asz conjecture, Com-binatorica 12 (1992), 155–160.12.
D. Larman and C. Rogers, The realization of distances within sets in Euclidean space,Mathematika 19 (1972), 1–24.13. D. Larman, Open problem 6, Convexity and Graph Theory (M. Rozenfeld and J.
Zaks,eds. ), Ann.
Discrete Math., vol. 20, North-Holland, Amsterdam and New York, 1984,p.
336.14. M. Lassak, An estimate concerning Borsuk’s partition problem, Bull.
Acad. Polon.
Sci. Ser.Math.
30 (1982),449–451.15. C. A. Rogers, Some problems in the geometry of convex bodies, The Geometric Vein—TheCoxeter Festschrift (C. Davis, B. Gr¨unbaum, and F. A. Sherk, eds.
), Springer-Verlag, NewYork, 1981, pp. 279–284.16.
O. Schramm, Illuminating sets of constant width, Mathematika 35 (1988), 180–199.17. M. Simonovits and V. S´os, Graph intersection theorems.
II, Combinatorics (A. Hajnal andV. S´os, eds.
), North-Holland, Amsterdam, 1978, pp. 1017–1030.
4JEFF KAHN AND GIL KALAIDepartment of Mathematics, Rutgers University, New Brunswick, New Jersey 08903E-mail address: jkahn@math.rutgers.eduInstitute of Mathematics, Hebrew University, Jerusalem 91904, IsraelE-mail address: kalai%humus.huji.ac.il@relay.cs.net
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