On the Frankl–Rödl theorem

The chromatic number of Rn$\mathbb {R}^{n}$ with multiple forbidden distances

“…In 1987, Frankl and Rödl [11,Theorem 1.18] proved that 𝜒 𝑘 (ℝ 𝑛 ) ∶= 𝜒 𝑘 (ℝ 𝑛 ; 1) grows exponentially with 𝑛 for any 𝑘. For 𝑚 = 1 and general 𝑘, the best lower and upper bounds, due to Sagdeev [28] and Prosanov [24], respectively, are…”

“…\end{equation*}$$In 1987, Frankl and Rödl [11, Theorem 1.18] proved that $$\chi _{k}(\mathbb {R}^{n}):=\overline{\chi }_{k}(\mathbb {R}^{n};1)$$ grows exponentially with n for any k . For $$m=1$$ and general k , the best lower and upper bounds, due to Sagdeev [28] and Prosanov [24], respectively, are $$$$\begin{equation} {\left(1+\frac{1}{2^{2^{k+4}}}+o(1)\right)}^{n}<\overline{\chi }_{k}{\left(\mathbb {R}^{n};1\right)}\leqslant {\left(1+\sqrt {\frac{2(k+1)}{k}}+o(1)\right)}^{n}. \end{equation}$$$$For $$k=2$$, the case of an equilateral triangle, better quantitative bounds are known [22] using the slice rank method developed in [6, 8, 30].…”

“…The right‐hand side of (1.7) is nontrivial only when $$m+1>\Gamma _{\chi }^{-2}k$$, however the method used yields a nontrivial result for $$m\geqslant k$$ (as stated in Theorem 2 above). For k larger than m , the best lower bound comes from the one distance case due to Sagdeev [28].…”

The Chromatic Number of $\mathbb{R}^{n}$ with Multiple Forbidden Distances

“…In 1987 Frankl and Rodl [12,Theorem 1.18] proved that χ k (R n ) := χ k (R n ; 1) grows exponentially with n for any k. For m = 1 and general k the best lower and upper bounds, due Sagdeev [31] and Prosanov [27], respectively, are…”

“…This is because previous methods have been based on Frankl and Rodl's approach [12,Theorem 1.18], which inductively applies a result for the k = 1 case. The right hand side of (1.7) is nontrivial only when m + 1 > Γ −2 χ k, however the method used yields a nontrivial result for m ≥ k. Outside of this range, for k larger than m, the best lower bound comes from the one distance case due to Sagdeev [31].…”