Path connected coloured sets on the squared paper

I believe that such a 3x3 square does not necessarily exist.

A counterexample would take the form of an infinite still life pattern in the life-like cellular automaton rule B123678/S34 (these rules are chosen so that the only patterns that remain stable are the ones in which the number of live cells in each 3x3 box is 4 or 5). Additionally, both the live and dead cells of the pattern should be connected.

But as the following partial double spiral shows (copy and paste it into Golly to view and test) it's possible to form partial double-spiral patterns that, at least in the center of the pattern, have the desired properties. I don't see any good reason why it shouldn't be possible to continue the spiral infinitely.

x = 31, y = 31, rule = B123678/S34
14b4o$12b3o2b3o$10b3o6b3o$8b3o3b4o3b3o$6b3o3b3o2b3o3b3o$5b2o3b3o6b3o3b
2o$5bo2b3o3b4o3b3o2bo$4b2ob2o3b3o2b3o3b2ob2o$4bo2bo2b3o6b3o2bo2bo$3b2o
b2ob2o3b4o3b2ob2ob2o$3bo2bo2bo2b3o2b3o2bo2bo2bo$2b2ob2ob2ob2o6b2ob2ob
2ob2o$2bo2bo2b2obo2b4o2bo2bo2bo2bo$2bo2bo2bo2bob2o2b2ob2ob2ob2ob2o$b2o
b2ob2ob2o2bo2bo2bo2bo2bo2bo$b2ob2ob2ob2ob2ob2ob2ob2ob2ob2o$bo2bo2bo2bo
2bo2bo2bo2bo2bo2bo$2ob2ob2ob2ob2ob2o2bo2bo2bo2bo$o2bo2bo2bo2b2o2bob2ob
2ob2ob2o$2ob2ob2ob2o4b2obo2bo2bo2bo$bo2bo2bo2b6o2bo2bo2bo2bo$b2ob2ob2o
3b2o3b2ob2ob2ob2o$2bo2bo2b3o4b3o2bo2bo2bo$2b2ob2o3b6o3b2ob2ob2o$3bo2b
3o3b2o3b3o2bo2bo$3b2o3b3o4b3o3b2ob2o$4b3o3b6o3b3o2bo$6b3o3b2o3b3o3b2o$
8b3o4b3o3b3o$10b6o3b3o$12b2o3b3o!

Here's a screenshot:

_{(source: uci.edu)}

I think the result is false. Consider a sequence of drawings, one of which I will represent here:

&&&&&&&&&&&&&&&
&  &  &  &  &
&  &  &  &  &
&& && && && &&
&  &  &  &  &
&  &  &  &  &
&& && && && &&
&& && && && &&

This is a coloring of a 9 x 15 region which satisfies the conditions and has no 3x3 square with six unit squares of the same color. (unfortunately, there are some rendering problems as I am not seeing how to control the line spacing.) It should be clear how to extend this for mxn regions in which both m and n are arbitrarily large. Now the idea is to develop a compactness style argument which expresses connectedness of both regions, the lack of a 3x3 subregion with at least 6 squares of one color, and the arbitrary size of the diagram. While I do not have the argument nailed down, I suspect one can use this to show an infinite domain colored in such a way as to preserve all the properties. This (plus other poster's evidence to the contrary) is why I believe the poster's assertion that such a 3x3 square exists that contains at least 6 squares of one color is false.

Gerhard "Ask Me About System Design" Paseman, 2010.09.05