Timeline for a block design question: Does every special 1-design admit a partition which respects enough of the blocks?
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| Dec 18, 2013 at 18:40 | comment | added | Gerhard Paseman | For example, if n is 2 mod 3, then only two "adjacent exposed" elements of a ring must remain uncovered by a 3-refinement restricted to 3-sets inside that ring. Now arrange some "transverse" blocks that mess with this adjacent property. Gerhard "You Can Finish It Up" Paseman, 2013.12.18 | |
| Dec 18, 2013 at 18:33 | comment | added | Gerhard Paseman | No, I don't have one. However, I would start with the following "gadget" in trying to build one. Arrange n>4 elements in a ring, and cover adjacent elements with 4-sets. Remove one of the 4-sets and call this a ring. (Later you will add elements and blocks to cover the 4 "exposed" elements with lambda = 3.) Refining this by a three-set partition limits the possibilities of which of the exposed elements are uncovered, and $n$ gives you some control. Try putting some of these rings and a few blocks together. Gerhard "Does That Cover It Now?" Paseman, 2013.12.18 | |
| Dec 18, 2013 at 9:45 | comment | added | j.s. | Gerhard, thanks. In fact, I assume D is a connected. Can you give a counterexample in this case? | |
| Dec 17, 2013 at 20:11 | history | edited | Gerhard Paseman | CC BY-SA 3.0 |
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| Dec 17, 2013 at 20:05 | history | answered | Gerhard Paseman | CC BY-SA 3.0 |