We can ask these questions also for equiped with Nets Katz’ addition (see link below) which is an intermidiate version of addition for p-ary integers between the “nim” addition (addition in ) and ordinary addition (of q-ary numbers) over the integers.
https://gilkalai.wordpress.com/2016/05/17/polymath-10-emergency-post-5-the-erdos-szemeredi-sunflower-conjecture-is-now-proven/#comment-24602

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