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A METRIZATION for POWER-SETS and CARTESIAN PRODUCTS TOTH APPLICATIONS to COMBINATORIAL ANALYSIS DISSERTATION Presented in Partia

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A METRIZATION for POWER-SETS and CARTESIAN PRODUCTS TOTH APPLICATIONS to COMBINATORIAL ANALYSIS DISSERTATION Presented in Partia A METRIZATION FOR POWER-SETS AND CARTESIAN PRODUCTS TOTH APPLICATIONS TO COMBINATORIAL ANALYSIS DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of The Ohio State University ROBERT SILVERMAN, B. Sc., M. A. ****** The Ohio .State University 1958 Approved by Adviser Department of Mathematics Acknowledgements ®he author wishes to express his sincere appreciation to Professors Marshall Hall, Jr. and Henry 5. Mann for their reading of the dissertation and their suggestions, and especially to his adviser, Professor D. Hansom Whitney, to Professor Herbert J. Terser, and to his wife, Donna, for their continued encouragement and assistance. ii Contents Section Page 1. Introduction ..................................... 1 2. Abstract metric spaces ........................ 1 3. A metrization for power-sets and Cartesian products .• U U> The metric space 5^(n) ........ 5 5. Metric characterizations of some classical configurations and their generalizations ........... 6 6. Some theorems for the metric space S^(n) ........... 18 7. Future research .......................... 31 Bibliography......................... 33 Autobiography......................................... 35 iii 1. Introduction. Virtually every combinatorial configuration may be phrased in terns of certain conditions on families of subclasses of the power-set Ug (the class of all subsets of S) of some abstract set S. In view of this, it would seem worthwhile to investigate systematically some aspects of the behavior of Ug. In this paper a method of structuring the power-set is proposed /by defining a distance function over Ug. Since Cartesian products may be viewed as subsets of power-sets, this definition also leads to a metrization for such sets. The present paper is concerned primarily with an examination of some of the metric properties of Cartesian products and their relation to some of the classical configurations and their generalizations. 2. Abstract metric spaces. In order to reduce to a minimum the introduction of new terminology, wherever feasible the author has adopted that used by ELumenthal [Ul [f>l, whose excellent books also nourished several interesting trains of thought. DEFINITION 1. Given a set M, an r-dimensional metric over M, r > 2, is a mapping which associates with every ordered r-tuple a]_,a2, ...,ar of elements of M, a non-negative real number d(a^,...,ar), termed the distance of the points a^, ...,ar, satisfying: M l . d( a^,..., &p) > 0.* M 2. d(a-j_, ...,ar) = 0 if and only if a-^ = a2 s ... = ar. 1 2 M 3 . (Symmetry) If • • .>ir is any permutation of the integers 1,2,..., r, then d( ap ..., ar) = d(a^, . a ^ ) . M U« (Triangle inequality) For every set of r + 1 points ) «• •» aj. + ]_ of M, d(a^ *«• >ar) < ? d(a^, ...,a^ _ + ^>..»,ar + ^)» The postulate M 2 differs somewhat from that in ELumenthal [U, P* 126], but is more suitable to our purposes. Also, in keeping with the progressive spirit of the times, our "dimension” is one higher than the comparable "dimension” in ELumenthal. In the applications there frequently will be metrics of various dimensions defined over M with connecting conditions. Generally we will use the symbol M to denote either the set M, or the metric space consisting of M and the metric d over M. Wien it is desirable to differentiate explicitly the two concepts, we will symbolize the latter by (M,d). We now define some concepts for conventional (two-dimensional) metric spaces. In most instances the extension to r-dimensional spaces is obvious. In the following, M denotes a metric space, and E a subspace of M. DEFINITION 2. If M ■ £a-^,a2, ...j is countable, it may be completely specified by the symmetric distance matrix A - (ai;j)> ai;j ■ d(a±,a3). 3 DEFINITION 3. For a in M, r > 0, the open sphere aid closed sphere with center a, radius r are defined, respectively, by Note that in general a sphere need not have a unique center or radius. DEFINITION li. The major diameter A(E), of E, and the minor diameter /( E), of E are defined by A(E) ■ l.u.b. d(x,y) for x,y in E, /(E) = g.l.b. d(x,y) for x,y in E, x y. If E contains fewer than two points, define /(E) = 0. DEFINITION E>. The t-extent of E, e(E,t), is the greatest integer m such that E contains m distinct points with minor diameter greater than t. If no two points of E have distance greater than t, set e(E,t) * 1, while if for n arbitrarily large there are n points of E with minor diameter exceeding t, define e(E,t) = + °o. DEFINITION 6. Two metric spaces M and M' are isometric provided there exists a mapping a from M onto M' such that d(x,y) » d(a(x),a(y)) for all x,y in M. We write M * M'. Note that a is biunique since a(x) ■ a(y) implies d(a(x),a(y)) * d(x,y) “ 0. If M = M', the isometry is termed a motion. Two subsets of M are superimposable provided a motion exists that maps one onto the other. u DEFINITION 7. E is a metric basis of M provided each point of M is uniquely deteimined by its distances from the points of E. The basis is irreducible provided no proper subset of E is a metric basis, and is minimal provided M has no metiic basis of smaller cardinality. Finally, the basis is complete provided for each subset E' of M with E “ E', and each subset T' of M containing E", a subset T of M exists such that the isametxy of E with E' can be extended to T S T'. The extension is obviously unique. Also, note that for M finite, for completeness it is sufficient to require simply that E and E' be superimposable for each subset E' of M with E ~ E'. DEFINITION 8. For x in M, the distance of x from E is given ty d(x,E) » g.l.b. d(x,y) for y in E. 3. A metrization for power-sets and Cartesian products. Given a set S, let Ug denote its power set. For A in Ug, let N(A) be the number of elements in A, where + ®o is an admissible value. For A^, •••,Ar in Ug define: DEFINITION 9. d(A!,...,Ar) - N(\JA± - A A i ) / r . THEOREM 1. Under the above definition, d is an r-dimensional metric over Ug. Proof. M 1, M 2, M 3 are immediate. Let A^,...,Aj, + be in Ug, Let a L ° d(A]_, ...,Aj.), R ** 2 d(A]_, .• Aj_ «• i>Aj_ + x* • ••jAp + x^* <■'=/ 5 If R = + «o , then certainly L < R, so we may assume R < + oo . Let x be arbitrary in S. Then x can contribute 0 to R if and only if x is either in all of the r + 1sets or innone. In either case it also contributes 0 to L. In all other cases x contributes at least l/r to R and at most l/r to L. Hence L < R. Thus M U is satisfied. In particular, definition 9 defines an r-dimensional metric for Cartesian products as follows: Given the sets S , ...,,S , X Jx denote by n Si ° Si x S 2 x ... x Sjj ■ ^(s^, ...,sk)» s^ in their Cartesian product of ordered k-tuples. Then it may be regarded as a subclass of Ug(k ) x where 2, ...,kj, by regarding the element (a^,. .^a^) of n as the subset |(l,a1),...,(k,ak )j of S(k) x k* ^*1° metric space S^(n). Let S(n) denote a set with n elements, n > 1. Unless otherwise indicated, S(n) will be taken as the first n positive integers. We denote by S^-(n), k > 1, the k-fold Cartesian product of S(n), S ^ n ) - {(x l , ...,xk); X£ in S(n)j, and assume that S^(n) has been metrized as in the preceding section. Although it appears that higher dimensional metrics will be necessary to characterize such items as the theorem of Desargues, in the present paper we confine our attention to the two- dimensional metric, which for this special case we may restate: For x,y in S^n), x - ( x ^ . ^ ) , y - (y-p ...,yk ), d(x,y) is the number of subscripts i for which x^ / y^, i = 1, ...,k. For 0 < r < k, every set of nr + 1 elements of Sk (n) has minor diameter at most k-r. Hence the k-r extent of every subspace E of S^(n) satisfies e(E, k - r) < nr . We next define terms to describe subspaces which attain this maximum extent. DEFINITION 10. A subspace E of S^(n) is r-orthogonal, 0 < r < k, provided e(E, k -r) = nr. If in addition E contains precisely nr elements, E is termed an L(n,k,r) space. (Thus E is r-orthogonal if and only if E contains an L(n,k,r) space.) For a given subspace E, r-orthogonality does not imply (r-1) -orthogonality. S^(n) always has orthogonality 0,1 and k. Indeed, any point constitutes an L(n,k,0) space/ the points (i,i,...,i), i » 1,...,n comprise an L(n,k,l) space, and S^(n) is itself an L(n,k,k) space. For values between 1 and k the property becomes non-trivial, and, as we shall see in the following section, is related to some of the classical unsolved problems in combinatorial analysis.
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