Just to remark that the upper bound for , ascribed in 228. to Edel, is actually present in Meshulam’s paper, in the most explicit form.

]]>I tried a pattern of points based on Terry.39’s suggestion. It seems to give asymptotic results similar to what he thought.

First define a sequence, of all positive numbers which, in base 3, do not contain a 1. Add 1 to all multiples of 3 in this sequence. This sequence does not contain a length-3 arithmetic progression.

It starts 1,2,7,8,19,20,25,26,55, …

Second, list all the (abc) triples for which the larger two differ by a number

from the sequence, excluding the case when the smaller two differ by 1, but then including the case when (a,b,c) is a permutation of N/3+(-1,0,1)

This had numerical asymptotics for close to

$latex 1.2-\sqrt(\log(n)) between n=1000 and n=10000

Sune.273: Sorry, I haven’t written all the numbers in yet for my pattern – I am only up to 70, so another formula is used to give lower bounds after that.

]]>As this thread is beginning to get quite long, I am moving it to a new thread, starting at 700:

]]>Dear Thomas,

It unfortunately seems to me that the bottom row of the middle contains some horizontal lines (e.g. the three centres of the three ‘s in this bottom row). There are also some lines on the top row and on the left and right columns of this middle .

(If you write each of the ‘s in xyz notation, one sees the problem; there are a bunch of rows and columns which only have two of the three patterns x, y, z and so let some lines slip in. The ‘s with three red squares in them are the intersection of two of the x, y, z and can kill more lines, but unfortunately this doesn’t cover all the possible lines.)

But it could be that some permutation of this sort of strategy could work… it may provide the first good example we have which is not based off of the set.

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