Say the lowest possible density is :

1) 5 voters out of 29 – behaves like for

2) 7 voters — behaves like for some .

3) 1 voter (me) g(n) is larger than any power of log (n) but g(g(n)) is smaller than log n

4) 1 voter: g(n) is smaller than for any but g(g(n)) is larger than log n.

5) 6 voters: g(n) behaves like for some .

6) 9 voters g(n) behaves like exp (sqrt (log n))

The result by Schoen and Sisask strengthen the "case" for 5-6. (but the regularity consideration weakens this strengthening…) I am fond of the possibility that g(n) is larger than any power of log n but is still on the "polynomial side".

For the capset problem the results were very decisive on the exponential side (I still hope better examples can be found.)

]]>1) Out of 29 voters 5 thought that it would be n/(log n)^c for c1

3) 1 voter said n/log n^c is enough for any c but if the answer is n/g(n) g(n) is still on the polynomial side in the sense that it g(n) is smallet then some exp exp… exp log log log …cn where the number of logs is one more then the number of exps

4) 1 voter said that g(n) is larger on the “linear side” exp exp..exp log log log …log n with the same numbers of logs and exps (but not as large as the next answers)

5) 6 voters said exp. log n^c for some c< 1/2

6) 9 voters said exp log n^1/2

I was the voters in 3. For some fantastic wishful thinking baseless reasons

http://gilkalai.wordpress.com/2009/03/25/an-open-discussion-and-polls-around-roths-theorem/

]]>In this recent preprint, Schoen and Sisask obtain Behrend-shape upper bounds for 4 variable equations: http://arxiv.org/pdf/1408.2568v1.pdf

]]>Dick Gross refers to him as Jim several times during his introduction.

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*Thanks — I’ve corrected it now.*

Gerhard “Or Too Much Gender Publicity” Paseman, 2014.09.04

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