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Jun 23, 2018 at 7:31 history edited Aaron Meyerowitz CC BY-SA 4.0
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Jun 22, 2018 at 16:17 comment added Aaron Meyerowitz Yes, that is indeed pretty clear! There should be an easy proof based on that.
Jun 22, 2018 at 12:06 comment added user44191 As a note, the fact that all other lines contain an element of $S$ is pretty clear. Either they are horizontal or they contain an element of the first line; the first line is entirely nonzero, and none of the horizontal lines are zero.
Jun 22, 2018 at 6:08 comment added Seva I have yet to check carefully your construction, but I would not be surprised if there are examples of some general sort ($p>5$).
Jun 22, 2018 at 5:14 history answered Aaron Meyerowitz CC BY-SA 4.0