What can you say about numbers $d$ where $l$ is 1 or 2? Is it conceivable that the upper bound is constant? Gerhard "If You Dream, Dream Big" Paseman, 2015.04.16

The above can be improved, since what you care about are the triangle vertices being rainbow and not the edges. However, I think it is best if you leave Gamma out of the picture and talk instead about forbidden rainbow configurations of points. Gerhard "Simplify, But Not Too Much" Paseman, 2015.04.14

In spite of your efforts, it is not clear what you ask of the coloring. My wording: Given R^3, A,B,C,D on a rhombus as stated, and start a three coloring with the assigned colors to the four points, can the coloring be extended to all of R^3 so that no unit equilateral triangle is rainbow? In particular, we need at least three circles around three edges to receive only two colors. I think this formulation avoiding Gamma is clear, since you are not asking for a proper graph coloring (neighboring vertices get different colors). Gerhard "Gamma Free For More Clarity" Paseman, 2015.04.14

I'm going to spout some moderately informed nonsense, with the hope that some sense and a possible answer you need can be found in it. Such classes can be imagined as unions or coproducts of varieties, which is like saying a set is a union of its elements. If there is a uniform description to these elements, there may be more interest; otherwise, do you care about my laundry list and should I care about yours? Coproducts should be better understood, but that may mean doing a lot of laundry. Gerhard "Read This Metaphorically, Not Literally" Paseman, 2015.04.13

Welcome to MathOverflow! Perhaps they will make a "Human" badge for "edit on first post". Gerhard "Has Plenty Of Badges Already" Paseman, 2015.04.02

Consider that a lower bound derives from the number of partitions of n-1 (say) into d parts. I would expect exponential growth in n even for d approaching n/e. Gerhard "Much Less For Bigger D" Paseman, 2015.04.02

Try a coordinate system change: move $x_0$ to the origin and then use spherical coordinates. See if you can use some analysis to find an optimal r. Gerhard "Maybe Cylindrical Coordinates Will Work" Paseman, 2015.04.01

I gave out some interesting challenge problems to my TA section for Linear Algebra/Differential Equations. One undergraduate student, Roger House, pushed back with a question of his own. The question and answer weren't original, but they were new to me and his further work on his question inspired mathoverflow.net/a/11902 as well as a result of George Bergman (the class professor) on splitting certain binary sets. It also spurred my first independent results in combinatorics. Gerhard "Yes, Those Were The Days" Paseman, 2015.04.01

An instance where transseries wouldn't have helped, I suppose. Gerhard "Guess Geometry's Good For Something" Paseman, 2015.04.01

I have seen that. That is one chapter out of Volume II, which has between 6 and 9 chapters. As far as I know, the actual volume II is still being rewritten. Gerhard "Has Old Volume Two Draft" Paseman, 2015.03.27

Ah, Emil Jeřábek has appeared. He is in recent touch with universal algebra; my knowledge is rusty and gappy. I'll let him take over. Gerhard "Ask Me About Gappy Math" Paseman, 2015.03.23

OK. Based on the definition, you have either clone equivalence, or some form of clone bi-embeddability. Certainly note that the function phi would extend to embedding the clones of each algebra into the other, but you need to check for projections and that composition is preserved. I don't recall which of the basic universal algebra texts have sections on interpretability: try Burris-Sankappannavar and McKenzie-McNulty-Taylor. Gerhard "Now Knows About Rational Equivalence" Paseman, 2015.03.23

You would have to provide more of the definition of rational equivalence. It sounds though like bi-interpretability might be closest. I'm guessing that you essentially want terms in each language so that each basic operation in the other language is given by such terms. Sort of like isomorphism of clones. My example is Boolean rings versus Boolean algebras. Gerhard "Doesn't Know About Rational Equivalence" Paseman, 2015.03.23

Consider a translation-only tiling by a marked triomino, marked with 1 and 2 and 3, say a 3x1 tile, and mark the occurrences where 1, 2, or 3 tile occur. This induces a pattern of numbers up to translation. How many 1,2,3 paintings of the plane are there from translation only tessellations? (Uncountably many, I think.) Can one pass from such a painting to a decomposition in similar tiles? Gerhard "Look At It From Underneath" Paseman, 2015.03.20