Embedding of a pseudo-point residual design into a Möbius plane

• Published in: Discrete mathematics Vol. 37; no. 1; pp. 29 - 33
• Main Author: Chan, Agnes Hui
• Format: Journal Article
• Language: English
• Published: Elsevier B.V 1981
• ISSN: 0012-365X; 1872-681X
• DOI: 10.1016/0012-365X(81)90137-0

Let U be a class of subsets of a finite set X. Elements of U are called blocks. Let υ, t, λ and k be nonnegative integers such that υ⩾ k⩾ t⩾0. A pair ( X, U ) is called a ( υ, k, λ) t- design, denoted by S λ( t, k, υ), if (1) | X| = υ, (2) every t-subset of X is contained in exactly λ blocks and (3) for every block A in U , | A| = k. A Möbius plane M is an S 1(3, q+1, q 2+1) where q is a positive integer. Let ∞ be a fixed point in M. If ∞ is deleted from M, together with all the blocks containing ∞, then we obtain a point-residual design M *. It can be easily checked that M * is an S q (2, q+1, q 2). Any S q (2, q+1, q 2) is called a pseudo-point-residual design of order q, abbreviated by PPRD( q). Let A and B be two blocks in a PPRD( q) M *. A and B are said to be tangent to each other at z if and only if A∩ B={ z}. M * is said to have the Tangency Property if for any block A in M *, and points x and y such that xϵ A and y∉ A, there exists at most one block containing y and tangent to A at x. This paper proves that any PPRD( q) M * is uniquely embeddable into a Möbius plane if and only if M * satisfies the Tangency Property.