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Radó proved that this number is at least 1/9 and conjectured that it is at least 1/4 a constant which cannot be further improved. This assertion was proved for the case of equal squares independently by A. Sokolin, R. Rado, and
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of a unit interval, one can select a subset consisting of pairwise disjoint intervals with total length at least 1/2 and that this number cannot be improved. He then asked for an analogous statement in the plane.
208:
306:
433:
241:, much work was devoted to establishing upper and lower bounds in various classes of shapes. By considering only families consisting of sets that are parallel and congruent to
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Problems analogous to Tibor Radó's conjecture but involving other shapes were considered by
Richard Rado starting in late 1940s. A typical setting is a finite family of
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If the area of the union of a finite set of squares in the plane with parallel sides is one, what is the guaranteed maximum total area of a pairwise disjoint subset?
75:
Radó's conjecture, by constructing a system of squares of two different sizes for which any subsystem consisting of disjoint squares covers the area at most
642:
704:
537:
513:
124:
383:{\displaystyle 0.1179\approx {\frac {1}{8.4797}}\leq F({\textrm {square}})\leq {\frac {1}{4}}-{\frac {1}{384}}\approx 0.2474,}
229:, i.e. consist of disjoint sets, and bars denote the total volume (or area, in the plane case). Although the exact value of
699:
558:
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Bulletin de l'Académie
Polonaise des Sciences, Série des Sciences Mathématiques, Astronomiques et Physiques
300:) that improve upon earlier results of R. Rado and V. A. Zalgaller. In particular, they proved that
107:
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In 2008, Sergey Bereg, Adrian
Dumitrescu, and Minghui Jiang established new bounds for various
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Bereg, Sergey; Dumitrescu, Adrian; Jiang, Minghui (2010), "On covering problems of Rado",
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673:, Matematicheskoe Prosveshchenie, Ser. 2 (in Russian), vol. 5, pp. 141–148,
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253:), which turned out to be much easier to study. Thus, R. Rado proved that if
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and has been generalized to more general shapes and higher dimensions by
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concerning covering planar sets by squares. It was formulated in 1928 by
21:
270:
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ranges over finite families just described, and for a given family
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666:
524:, Problem Books in Mathematics, New York: Springer-Verlag,
203:{\displaystyle F(X)=\inf _{S}\sup _{I}{\frac {|I|}{|S|}},}
106:, for example, a square as in the original question, a
612:—— (1951), "Some covering theorems (II)",
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640:
Sokolin, A. (1940), "Concerning a problem of Radó",
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671:Математика, ее преподавание, приложения и история
559:"Sur un problème relatif à un théorème de Vitali"
452:(1973), "The solution of a problem of T. Radó",
154:
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614:Proceedings of the London Mathematical Society
588:Proceedings of the London Mathematical Society
237:) is not known for any two-dimensional convex
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79:of the total area covered by the system.
428:{\displaystyle f(X)\geq {\frac {1}{6}}}
586:(1949), "Some covering theorems (I)",
225:ranges over all subfamilies that are
77:1/4 − 1/1728 ≈ 0.2494
48:, Tibor Radó observed that for every
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44:, motivated by some results of
497:; preliminary announcement in
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705:Unsolved problems in geometry
522:Unsolved Problems in Geometry
507:10.1007/978-3-540-69903-3_27
721:
20:is an unsolved problem in
667:"Замечания о задаче Радо"
530:10.1007/978-1-4612-0963-8
486:10.1007/s00453-009-9298-z
269:is a centrally symmetric
626:10.1112/plms/s2-53.4.243
600:10.1112/plms/s2-51.3.232
265:) is exactly 1/6 and if
245:, one similarly defines
18:covering problem of Rado
564:Fundamenta Mathematicae
435:for any convex planar
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384:
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83:Upper and lower bounds
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385:
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514:Falconer, Kenneth J.
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68:. However, in 1973,
512:Croft, Hallard T.;
281:) is equal to 1/4.
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700:Discrete geometry
616:, Second Series,
590:, Second Series,
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42:Wacław Sierpiński
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663:Zalgaller, V. A.
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518:Guy, Richard K.
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257:is a triangle,
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93:Euclidean space
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66:V. A. Zalgaller
46:Giuseppe Vitali
40:In a letter to
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89:convex figures
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70:Miklós Ajtai
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30:Richard Rado
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650:: 871–872,
620:: 243–267,
594:: 232–264,
571:: 228–229,
555:Radó, Tibor
227:independent
102:to a given
36:Formulation
689:Categories
679:0145.19203
656:0023.11203
577:54.0098.02
443:References
100:homothetic
26:Tibor Radó
499:SWAT 2008
460:: 61–63,
413:≥
393:and that
372:≈
359:−
346:≤
327:≤
314:≈
98:that are
73:disproved
665:(1960),
557:(1928),
520:(1991),
50:covering
22:geometry
634:0042149
608:0030782
548:1107516
494:2609053
466:0319053
271:hexagon
110:, or a
91:in the
677:
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375:0.2474
338:square
322:8.4797
311:0.1179
292:) and
213:where
118:. Let
534:ISBN
116:cube
108:disk
16:The
675:Zbl
652:Zbl
622:doi
596:doi
573:JFM
526:doi
503:doi
482:doi
367:384
155:sup
145:inf
691::
669:,
648:26
646:,
630:MR
628:,
618:53
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439:.
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104:X
96:R
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