Timeline for Is the domination number of a combinatorial design determined by the design parameters?
Current License: CC BY-SA 3.0
11 events
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| Mar 9, 2014 at 17:00 | history | edited | Peter Dukes | CC BY-SA 3.0 |
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| Mar 9, 2014 at 14:07 | comment | added | Dima Pasechnik | GAP package Design can apparently produce such a list for you, see gap-system.org/Manuals/pkg/design/htm/CHAP007.htm | |
| Mar 9, 2014 at 11:51 | comment | added | Peter Dukes | Oh, I am happy to e-mail you the 80 systems sometime. | |
| Mar 9, 2014 at 11:49 | comment | added | Peter Dukes | Sorry about the confusion. My added comment just says that if your search fails for $(15,3,1)$-designs, then you can get a counterexample for $(15,3,13)$-designs. On the one hand, 13 copies of the geometry results in domination number 11, while the complete design (all $\binom{15}{3}$ triples) has domination number $12+1 = 13$. If you don't like repeated blocks, or $\lambda=13$ is too high, I suspect you can apply the same sort of idea with $\lambda=3$ and permuting around the copies of $PG_3(2)$. | |
| Mar 9, 2014 at 11:44 | history | edited | Peter Dukes | CC BY-SA 3.0 |
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| Mar 9, 2014 at 11:42 | comment | added | Felix Goldberg | I am starting to consider Steiner systems so will try to check your argument on some examples. P.S. Is there a place where one can obtain all 80 systems in a machine format? I only have the 44 listed by Spence on his excellent webpage maths.gla.ac.uk/~es/bibd/15-3-1 | |
| Mar 9, 2014 at 11:39 | comment | added | Felix Goldberg | I think I lost you in the extra comment. :( | |
| Mar 9, 2014 at 11:22 | comment | added | Peter Dukes | Attacking my own reasoning: it does seem plausible that, in each system, some 8-subset could generate all but three blocks. Anyway I think I have a "cheat" solution, based on the above. If we take 13 copies of $PG_3(2)$ on the same points, the domination number is still (at worst) 11 via a subspace plus four blocks. On the other hand, the complete design of all triples on 15 points has domination number 13, I believe. So the answer is negative for (15,3,13) designs (when repeated blocks are permitted). | |
| Mar 9, 2014 at 10:32 | history | edited | Peter Dukes | CC BY-SA 3.0 |
edited to weaken my claim and increase my worry somewhat on the disclaimer
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| Mar 9, 2014 at 10:09 | history | edited | Peter Dukes | CC BY-SA 3.0 |
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| Mar 9, 2014 at 9:54 | history | answered | Peter Dukes | CC BY-SA 3.0 |