• No results found

On the number of blocks in a generalized Steiner system

N/A
N/A
Protected

Academic year: 2021

Share "On the number of blocks in a generalized Steiner system"

Copied!
3
0
0

Bezig met laden.... (Bekijk nu de volledige tekst)

Hele tekst

(1)

File: DISTIL 280901 . By:DS . Date:09:07:01 . Time:06:52 LOP8M. V8.0. Page 01:01 Codes: 3503 Signs: 1767 . Length: 50 pic 3 pts, 212 mm

Journal of Combinatorial Theory, Series A  TA2809

journal of combinatorial theory,Series A 80, 353355 (1997)

NOTE

On the Number of Blocks in a

Generalized Steiner System

J. H. van Lint

Department of Mathematics, Eindhoven University of Technology, Eindhoven, the Netherlands Communicated by the Managing Editors

Received April 24, 1997

We consider t-designs with *=1 (generalized Steiner systems) for which the block size is not necessarily constant. An inequality for the number of blocks is derived. For t=2, this inequality is the well known De BruijnErdos inequality. For t>2 it has the same order of magnitude as the WilsonPetrenjuk inequality for Steiner systems with constant block size. The point of this note is that the inequality is very easy to derive and does not seem to be known. A stronger inequality was derived in 1969 by Woodall (J. London Math. Soc. (2) 1, 509519), but it requires Lagrange multipliers in the proof.  1997 Academic Press

We consider a so-called generalized Steiner system t&(n, V, 1), i.e., a collection B of subsets (blocks) of an n-set P (of n points) with the property that every t-subset of P is contained in exactly one block in B.

We represent such a system by a (0,1)-matrix A of size b by n, with b := |B|, where the i th row of A is the characteristic function of the i th block Bi# B.

A generalized Steiner system is called trivial if |B| =1.

Definition. We define ;

t, n to be the minimal number of blocks in a

nontrivial system t&(n, V, 1).

Theorem. For t2 we have

;t, n( ;t, n&1)t

\

n t

+

.

Proof. The proof is by induction. The case t=2 is the well known Erdos-De Bruijn inequality (if t=2 and |B| >1, then |B| n; cf. [2, article no.TA972809

353

0097-316597 25.00

Copyright  1997 by Academic Press All rights of reproduction in any form reserved.

(2)

File: DISTIL 280902 . By:DS . Date:09:07:01 . Time:06:52 LOP8M. V8.0. Page 01:01 Codes: 2532 Signs: 1801 . Length: 45 pic 0 pts, 190 mm

Theorem 19.1]). Now, assume that the theorem has been proved for t&1 and all n. Consider the matrix A of a t&(n, V, 1). If any column of A has only one 1, then the system is trivial. So, for any point, the derived design with respect to this point is a non-trivial (t&1)&(n&1, V, 1) system. This implies that all columns of A have at least ;t&1, n&1 ones. We now count

pairs (ai, k, aj, k) equal to (1, 1) with 1i<jb. By first choosing the pair

(i, j), we find at most t&1 such pairs (1, 1). So, the total number is at most

(t&1)

\

b 2

+

. In any column of A, we find at least

\

;t&1, n&1

2

+

such pairs. It follows that

(t&1)

\

;t, n 2

+

n

\

;t&1, n&1 2

+

 n(t&1) 2

\

n&1 t&1

+

, i.e., ;t, n(;t, n&1)t

\

n t

+

. K

Remark. Note that the t-subsets of a (t+set form a t&(t+1, V, 1)-system for which equality holds in the theorem.

In general this bound is weak. However, the result is easy to derive and certainly deserves to be an exercise in combinatorics books. If we fix t, the inequality is a diophantine equation in ; and n which probably has very few solutions. So equality is not to be expected except for the case already mentioned. For t=3 and n=5 we find ;( ;&1)30, where ;=6 would give equality. However, it is easy to see that a 3&(5, V, 1) design with 6 blocks does not exist. For t=3, we do find the interesting fact that ; grows like n32

. If t=3 and n=8, we find that ;>13.47, so a design with these parameters must have at least 14 blocks. Indeed, there is a S(3, 4, 8) with 14 blocks.

We now compare the result with the well known WilsonPetrenjuk inequality ( ;(n

s) if t=2s; cf. [2, Theorem 19.8]). Clearly, Wilson

Petrenjuk is stronger. For example, for t=4 it yields as the right-hand side in our inequality 6(n+1

4 ) instead of 4( n

4). Note, however, that in both

bounds the rate of growth of the bound for ; (for fixed t) is as nt2.

(3)

File: DISTIL 280903 . By:DS . Date:09:07:01 . Time:06:55 LOP8M. V8.0. Page 01:01 Codes: 2208 Signs: 1172 . Length: 45 pic 0 pts, 190 mm

We also compare our bound with a result due to Woodall (cf. [3]). This states that

;

\

n t

+<\

k t

+

, where k is the larger root of

(k&t+2)(k&t+1)=n&t+1

and the binomial coefficient is interpreted in the usual way if k is not an integer. For t=2, this is again the De BruijnErdos inequality. This bound is more difficult to derive but it is stronger than our simple inequality. In our example (t=3, n=8) it yields ;14 which is exact. For t=3 and n=5 it yields ;>6.05 showing the nonexistence of the 3&(5, V, 1) men-tioned above.

We remark that for Steiner systems with t=4 and n>23, Woodall's bound is larger than the WilsonPetrenjuk bound, showing that for these parameters a tight design cannot exist.

ACKNOWLEDGMENTS

The author thanks A. E. Brouwer and H. D. L. Hollmann for some very useful comments that led to the analysis and comparison.

REFERENCES

1. P. J. Cameron and J. H. van Lint, ``Designs, Graphs, Codes and their Links,'' Cambridge Univ. Press, Cambridge, 1991.

2. J. H. van Lint and R. M. Wilson, ``A Course in Combinatorics,'' Cambridge Univ. Press, Cambridge, 1992.

3. D. R. Woodall, The *&+ problem, J. London Math. Soc. (2) 1 (1969), 509519. 355 NOTE

Referenties

GERELATEERDE DOCUMENTEN

MIDTERM COMPLEX FUNCTIONS APRIL 20 2011, 9:00-12:00.. • Put your name and studentnummer on every sheet you

4.4 Garden traits identified as the most important predictors of the abundance (a-d) and species richness (e-f) of nectarivorous birds in gardens in Cape Town, South Africa.. All

Ter hoogte van dit profiel werd ook geboord om de dikte van het middeleeuws pakket en de hoogte van het natuurlijke pleistocene zandpakket te bepalen.. Het pleistoceen werd

• A submitted manuscript is the version of the article upon submission and before peer-review. There can be important differences between the submitted version and the

The algonthm analyzed m this paper is a vanant of the class group relations method, which makes use of class groups of bmary quadratic forms of negative discnminant This algonthm

Although we are able to prove that the cost shares computed by our algorithm are 2-budget balanced, they do not correspond to a feasible dual solution for any of the known

A tight lower bound for convexly independent subsets of the Minkowski sums of planar point sets.. Citation for published

As the aim of this simple model is to gain insight a highly simplified situation is considered as shown in figure 6 where we just consider three wells and no reservoir tank, we