Chromatic numbers of spaces with forbidden monochromatic triangles

New Lower Bounds for the Chromatic Number of a Space with Forbidden Isosceles Triangles

“…In Table 1, numerical results for Theorem 1 are given. In [29], an analogous table is presented as well. However, here we give a larger list of bounds.…”

“…Our choice is explained by maximizing the base of the exponential function that estimates the cardinality of the constructed collection. The optimality of the constant C 1 is due to a theorem by Ahlswede and Khachatrian (see [1][2][3][29][30][31]).…”

“…However, there is a detail related to the estimation of d: in Subsec. 4.4, we referred to [29], where, in a similar setting, it is proved that the asymptotic behavior of the logarithm of d is defined by the first summand in the outer sum, but this argument is no longer valid. That is why the statement of the theorem has to include the quantity C 3 (x, y) which, multiplied by C 2 (x), defines the coefficient of the first power of n in the asymptotic expression for the logarithm of the summand in the repeated sum which is obtained for j ∼ yn, i ∼ xn, n → ∞.…”