Isosceles Sets Yury Ionin October 10, 2015 1 2 P. Erd os, E 735, - PDF document

Isosceles Sets Yury Ionin October 10, 2015 1

2 P. Erd˝ os, E 735, The American Mathematical Monthly (1946). Six points can be arranged in the plane so that all triangles formed by triples of these points are isosceles. Show that seven points in the plane cannot be so arranged. What is the least number of points in the space which cannot be so arranged? P. Erd˝ os and L.M. Kelly (1947). Example . A set consisting of the center and two opposite points (poles) of a sphere and of the vertices of a regular pen- tagon inscribed in the equator is an isosceles set. Conjecture . There is no isosceles set of cardinality 9 in E 3 .

3 H. T. Croft (1962), 9-point and 7-point configurations in 3- space, Proc. London Math. Soc. (3) , 12 , 400–424. H. Kido (2006). Classification of isosceles eight-point sets in three-dimensional Euclidean space, Europ. J. Combin. , 27 , 329–341. Definition. A set X in a metric space with the distance function d is called isosceles if for any x, y, z ∈ X , at least two of the distances d ( x, y ) , d ( x, z ) , d ( y, z ) are equal. Definition. A set X in a metric space is called an s-distance set if there exist a set D of s positive real numbers such that the distance between any two distinct points of X belongs to D .

4 Any 2-distance set is isosceles.

5 The largest 1-distance set in E n is the set of n + 1 vertices of a regular n -simplex. E. Bannai, E. Bannai and D. Stanton (1983). An upper bound for the cardinality of an s -distance set in real Euclidean space. A. Blokhuis (1984). Few-distance sets. Theorem. The cardinality of an s -distance set in E n does � n + s � not exceed . s Known equality cases: n = 1 and ( n, s ) = (8 , 2). With each point p of an s -distance set S we associate a poly- nomial in n variables s ( || x − p || 2 − d 2 � f p ( x ) = i ) , i =1 where d 1 , . . . , d s are the distances between points of S . For p, q ∈ S , f p ( q ) = 0 if and only if p � = q .

6 The largest 2-distance set in E 1 consists of the endpoints and the midpoint of a line segment. The largest 2-distance set in E 2 consists of the vertices of a regular pentagon. There is no 2-distance set of 7 pts. in E 3 . (Croft, 1962) The 2-distance sets of cardinality 6 in E 3 are the vertices of a regular octahedron, the vertices of a 3-prism with regular faces, any set of six vertices of a regular icosahedron having no pair of opposite vertices. (Einhorn and Schoenberg, 1966) P. Lisonek (1997). New maximal two-distance sets. Found all 2-distance sets of maximal cardinality in E n for 4 ≤

7 n ≤ 7. The largest 2-distance set in E 4 consists of the midpoints of the ten edges of a regular 4-simplex.

8 Blokhuis’ Theorem (1984). The cardinality of an isosceles set in E n does not exceed � n +2 � . 2 Known cases of equality: n = 1, n = 2, and n = 8.

9 Blockhuis’ Lemma . If S is an isosceles set in a metric space M , | S | ≥ 2, then S can be split into disjoint subsets X and Y satisfying the following conditions: (i) X is a 2-distance set, | X | ≥ 2; (ii) every point of Y is a center of a sphere containing X . Moreover, if M = E n and Y is not empty, then the affine subspaces generated by X and Y are orthogonal and therefore dim S ≥ dim X + dim Y. (The dimension of a set of points in E n is the dimension of the least affine subspace containing this set.) Suppose the edges of a complete graph are colored in three or more colors. Suppose further that for any two vertices u and v and for any color c there is a path from u to v formed by edges of color c . Then the graph contains a triangle whose edges are

10 colored in three distinct colors. Let S be an isosceles sets in E 4 of the maximal cardinality. Then | S | ≥ 11. S = X ∪ Y , Y � = ∅ , dim X + dim Y ≤ 4. (i) dim X = 4, dim Y = 0 ⇒ | X | ≤ 10, | Y | = 1; (ii) dim X = 3, dim Y = 1 ⇒ | X | ≤ 6, | Y | ≤ 3; (iii) dim X = 2, dim Y = 2 ⇒ | X | ≤ 5, | Y | ≤ 6; (iv) dim X = 1, dim Y = 3 ⇒ | X | = 2, | Y | ≤ 8.

11 The binary Hamming space H n is the set of all binary words of length n with the distance between two words being the number of positions in which they differ. The words can be interpreted as vertices of the n -dimensional unit cube. The Hamming distance between two vertices is the square of the Euclidean distance: n n ( x i − y i ) 2 = || x − y || 2 . � � h ( x, y ) = | x i − y i | = i =1 i =1 An isosceles trialngle in H n is an isosceles triangle in E n .

12 Theorem (folklore). If S is a 1-distance set of maximum cardinality in H n , then | S | ≤ n + 1. Moreover, | S | = n + 1 if and only if there exists a Hadamard matrix of order n + 1. However, if dim S ≤ n − 1, then | S | ≤ n , and this bound is sharp for every n . Just take the intersection of H n and the hyperplane x 1 + · · · + x n = 1. Theorem (Delsarte (1975), Noda (1979)). � n +1 � If S is a 2-distance set in H n , then | X | ≤ + 1. 2 � n +1 � Moreover, if | S | = + 1, then 2 (i) n ≤ 2 and S = H n or (ii) n = 5 and S is the set of all points with the sum of coordi- nates even or the set of all points with the sum of coordinates odd.

13 Theorem (YI). If S is a 2-distance set in H n and dim S ≤ � n � n − 1, then | S | ≤ unless the following two conditions are 2 satisfied: (i) n ≤ 3 or n = 6 and (ii) S is contained in a face of the cube H n . Example . Let S be the intersection of H n and the hyperplane � n � x 1 + · · · + x n = 2. Then dim S = n − 1 and | S | = . 2 Blockhuis’ Lemma . If S is an isosceles set in a metric space M , | S | ≥ 2, then S can be split into nonempty disjoint subsets X and Y satisfying the following conditions: (i) X is a 2-distance set, | X | ≥ 2; (ii) every point of Y is a center of a sphere containing X .

14 The sphere of radius R and center 0 in H n is the intersection of H n and the hyperplane x 1 + · · · + x n = R .

15 Theorem (YI). If S is an isosceles set in H n but not a 2- � n � distance set, then | S | ≤ + 1. Moreover, if n = 6, then 2 | S | ≤ 12. Examples. If n = 5 or n ≥ 7, then let S consist of (1 , 1 , . . . , 1) and all vertices with exactly two nonzero coordinates. If n = 6, then let S consist of (0 , 0 , 0 , 0 , 0 , 0), (1 , 1 , 1 , 1 , 1 , 1) and all vertices with the first and two more coordinates equal to 1.