Embedding of a pseudo-point residual design into a Möbius plane
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)...
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Published in | Discrete mathematics Vol. 37; no. 1; pp. 29 - 33 |
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Main Author | |
Format | Journal Article |
Language | English |
Published |
Elsevier B.V
1981
|
Online Access | Get full text |
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Summary: | 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. |
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ISSN: | 0012-365X 1872-681X |
DOI: | 10.1016/0012-365X(81)90137-0 |