A celebrated but non-effective theorem of Tibor Gallai states that for any finite set $A$ of $\Z^n$ and for any finite number of colors $c$ there is a minimal $m$ such that no coloring of the finite $m^n$-grid can avoid that a homothetic image of $A$ is monochromatic. We find (or confirm) $m$ for equilateral triangles, squares, and various types of rectangles. Also, we extend the problem from homothety to general similarity, or to similarity generated using some special rotations. In particular, we compute Gallai similarity numbers for lattice rectangles similar to $1\times k$ (in all orientations) for $k=2,3,4$. The solutions have been found in the framework of the Satisfiability Problem in Propositional Logic (SAT). While some questions were solved using managed brute force, for the more computationally intensive questions we used modern SAT solvers together with symmetry breaking techniques. Some other minor questions are solved for triangles and squares, and new lower bounds are found for regular hexagons on the triangular lattice and for three-dimensional cubes in $\Z^3$.

翻译：Tibor Gallai的一个著名但非构造性定理指出：对于任意有限集合$A \subset \Z^n$及任意有限种颜色$c$，存在一个最小的$m$，使得对有限$m^n$-网格的任何着色都无法避免$A$的某个位似像成为单色的。本文确定（或验证了）等边三角形、正方形以及各类矩形的$m$值。此外，我们将问题从位似推广到一般相似变换，以及由特定旋转生成的相似变换。特别地，我们计算了与$1\times k$（所有方向）相似的格点矩形的Gallai相似数，其中$k=2,3,4$。这些解是在命题逻辑可满足性问题（SAT）的框架下得到的。部分问题通过受控的暴力搜索解决，而对计算量更大的问题，我们采用了现代SAT求解器结合对称性破缺技术。本文还解决了三角形和正方形的一些次要问题，并对三角格点上的正六边形及$\Z^3$中的三维立方体给出了新的下界。