http://michaelnielsen.org/polymath1/index.php?title=Probabilistic_formulation_of_Hadwiger-Nelson_problem#Lemma_39 ]]>

I think that your troubles with the grid are due to my lack of defining them, sorry for that. By grid, I mean , so the distances among parallel lines of the grid are all equal.

You’re right about the conditions of the origin, it can have any type of neighbors, just that we can use its bichromacity only for the particular set of neighbors in the (open) top halfplane having rational coordinates. So we can use this condition for any unit distance graph found so far by selecting an appropriate set of neighbors, but it seems to me that for these graphs we can’t win more than by simply using that two of its neighbors at distance are monochromatic, which we also know from e.g. probabilistic formulation (btw, the polymath wiki is down at the moment).

]]>When writing Roth, I wanted to use the result that in any coloring of the integers (with finitely many colors) we can find an arithmetic progression of length 3. This is a special case of van der Waerden, and in fact Roth proved something much stronger, instead of colorings about one dense set, which makes it a special case of Szemeredi, so probably I should have referred to my claim rather as van der Waerden. ]]>

To clarify my second question: I see your statement that a grid is â€˜niceâ€™ if thereâ€™s a non-trivial linear transformation that maps the grid to itself, but I’m not seeing how that is not vacuous. For, suppose you have a finite graph satisfying your convex-hull criterion but with arbitrary coordinates for the vertices. Then, surely you can draw horizontal and vertical lines through each upper-half-plane neighbour of the origin, and make your infinite grid just as copies of that, translated horizontally and vertically by all integer multiples of the distance between (respectively) the leftmost and rightmost gridlines and the topmost and bottom-most ones? Hence I think I must not be getting what you mean by a non-trivial linear transformation. I’m guessing that this is also something reliant on knowing what Gallai or Roth said.

Also, can I clarify something about your final point? You note that it’s OK for the origin to have neighbours in the lower half-plane, just that they can be blue. I’m guessing those neighbours can also have arbitrary coordinates, not necessarily on grid points – right? And similarly, can there also be additional neighbours of the origin in the upper half-plane that are not on grid points, just that they too are allowed to be blue? If so, I suspect we’re within reach, even if your answer to my “niceness” question is no, the grid has to be nicer than that.

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