We substantially improve a presently known explicit exponentially growing lower bound on the chromatic number of a Euclidean space with forbidden equilateral triangle. Furthermore, we improve an exponentially growing lower bound on the chromatic number of distance graphs with large girth. These refinements are obtained by improving known upper bounds on the product of cardinalities of two families of homogeneous subsets with one forbidden cross-intersection.

中文翻译：

---

改进的Frankl-Rödl定理及其某些几何后果

我们极大地改善了目前已知的具有禁止等边三角形的欧氏空间的色数上的指数增长的下界。此外，我们改进了具有大周长的距离图的色数上呈指数增长的下界。这些改进是通过改进已知的两个具有一个禁止交叉的同类子集的基数乘积的已知上限来实现的。