Timeline for Small maximal sets with no 3-AP?
Current License: CC BY-SA 3.0
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| Nov 6, 2011 at 16:25 | comment | added | Mike | Kevin, here's a link to the work that Odlyzko and Stanley did on greedy sequences: dtc.umn.edu/~odlyzko/unpublished Also, Odlyzko has some comments here on later work. There is also a couple of articles in Discrete Mathematics, one by Erods and coauthors, which are relevant. I've been thinking about this topic on and off over the past few days and may write up what little I have. | |
| Nov 6, 2011 at 2:50 | comment | added | Kevin O'Bryant | To which work(s) of Stanley and Odlyzko are you referring? | |
| Nov 5, 2011 at 7:40 | comment | added | Zack Wolske | Forgot about $2$ terms being on both ends of a progression, so really they may form $9$ progressions, and that's a bad example. A better choice is N=13, where there is a set of size 7 with no progressions, but the smallest maximal set is $'\\{3,5,8,10\\}'$, and this time a set of $3$ elements can't be maximal, by the above argument. | |
| Nov 5, 2011 at 7:25 | comment | added | Zack Wolske | An example of a set without 3-AP will be an upper bound for "smallest maximal subset", where maximal means no other numbers can be added without forming a 3-AP. For example, if $N=10$, then $\\{1,3,4,9,10\\}$ is a set with no 3-AP, and is also maximal, but it is not the smallest such set. $\\{2,3,5,6\\}$ has $4$ members, which is minimal, since $3$ elements can only form $6$ progressions (two for each pair), so at least one more element can be added. | |
| Nov 4, 2011 at 21:03 | history | asked | Mike | CC BY-SA 3.0 |