Abstract
In 1940 Fisher famously showed that if there exists a non-trivial (v,k,λ)-design, then λ(v-1)⩾k(k-1). Subsequently Bose gave an elegant alternative proof of Fisher’s result. Here, we show that the idea behind Bose’s proof can be generalised to obtain new bounds on the number of blocks in (v,k,λ)-coverings and -packings with λ(v-1)<k(k-1).
This is a preview of subscription content, log in via an institution to check access.
Similar content being viewed by others
References
I. Bluskov, M. Greig and M. K. Heinrich: Infinite classes of covering numbers, Canad. Math. Bull. 43 (2000), 385–396.
R. C. Bose: A Note on Fisher’s Inequality for Balanced Incomplete Block Designs, Ann. Math. Statistics 20 (1949), 619–620.
R. C. Bose and W. S. Connor: Combinatorial properties of group divisible incomplete block designs, Ann. Math. Stat. 23 (1952), 367–383.
D. Bryant, M. Buchanan, D. Horsley, B. Maenhaut and V. Scharaschkin: On the non-existence of pair covering designs with at least as many points as blocks, Combinatorica 31 (2011), 507–528.
Y. Caro and Zs. Tuza: Improved lower bounds on k-independence, J. Graph Theory 15 (1991), 99–107.
Y. Caro and R. Yuster: Packing graphs: the packing problem solved, Electron. J. Combin. 4 (1997), no 1, R1.
Y. Caro and R. Yuster: Covering graphs: the covering problem solved, J. Combin. Theory Ser. A 83 (1998), 273–282.
Y. M. Chee, C. J. Colbourn, A. C. H. Ling and R. M. Wilson: Covering and packing for pairs, J. Combin. Theory Ser. A 120 (2013), 1440–1449.
P. Erdős and H. Hanani: On a limit theorem in combinatorial analysis, Publ. Math. Debrecen 10 (1963), 10–13.
P. Erdős and A. Rènyi: On some combinatorical problems, Publ. Math. Debrecen 4 (1956), 398–405.
R. A. Fisher: An examination of the different possible solutions of a problem in incomplete blocks, Ann. Eugenics 10 (1940), 52–75.
Z. Füredi: Covering pairs by q 2+q+1 sets, J. Combin. Theory Ser. A 54 (1990), 248–271.
D. M. Gordon: La Jolla Covering Repository, http://www.ccrwest.org/cover.html.
D. M. Gordon and D. R. Stinson: Coverings, in The CRC Handbook of Combinatorial Designs, 2nd edition (Eds. C. J. Colbourn, J. H. Dinitz), CRC Press (2007), 365–373.
M. Greig, P. C. Li and G. H. J. van Rees: Covering designs on 13 blocks revisited, Util. Math. 70 (2006), 221–261.
S. M. Johnson: A new upper bound for error-correcting codes, IRE Trans. IT-8 (1962), 203–207.
P. Keevash: The existence of designs, arXiv:1401.3665.
W. H. Mills: Covering designs. I. Coverings by a small number of subsets, Ars Combin. 8 (1979), 199–315.
W. H. Mills and R. C. Mullin: Coverings and packings, in: Contemporary Design Theory, (Eds. J. H. Dinitz and D. R. Stinson), Wiley, (1992), 371–399.
V. Rödl: On a packing and covering problem, European J. Combin. 6 (1985), 69–78.
J. Schönheim: On coverings, Pacific J. Math. 14 (1964), 1405–1411.
D. R. Stinson, R. Wei and J. Yin: Packings, in: The CRC Handbook of Combinatorial Designs, 2nd edition (Eds. C. J. Colbourn, J. H. Dinitz), CRC Press (2007), 550–556.
O. Taussky: A recurring theorem on determinants, Amer. Math. Monthly 56 (1949), 672–676.
D. T. Todorov: Some coverings derived from finite planes, Colloq. Math. Soc. János Bolyai 37 (1984), 697–710.
D. T. Todorov: Lower bounds for coverings of pairs by large blocks, Combinatorica 9 (1989), 217–225.
Author information
Authors and Affiliations
Corresponding author
Rights and permissions
About this article
Cite this article
Horsley, D. Generalising Fisher’s inequality to coverings and packings. Combinatorica 37, 673–696 (2017). https://doi.org/10.1007/s00493-016-3326-9
Received:
Revised:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00493-016-3326-9