1 Correction to: Discrete Comput Geom (2018) 59:663–679 https://doi.org/10.1007/s00454-017-9936-1

In the following we refer to the original paper [1]. The main reason for writing this correction is the incorrect statement of Theorem 4. Section 4.1 should be corrected by replacing \([3]=\{1,2,3\}\), the set of three points, with \([4]=\{1,2,3,4\}\), the set of four points, everywhere. Moreover, the graph \(K_{3,3}\) is replaced by \(K_{4,4}\), and \(K_6\) with \(K_8\). Then the theorem is proved using a lemma analogous to the (corrected) Lemma 2, where the triple intersections are replaced with 4-wise intersections. The analogous lemma is proved by exactly the same method. A proof can also be found in [3] full version, Lemma 4. The resulting bound is indeed asymptotically worse than in the case of embedding:

$$\begin{aligned} g_d(n)=O\bigl (n^{d+1-4^{1-d}}\bigr ). \end{aligned}$$

The statement of Lemma 2 has to be corrected by assuming \(m\ge 3\) and setting \(t(S)=O\bigl (mn^{2/3}f^{{1}/{3}}(n)+n\bigr )\). The proof should also be modified by assuming \(k_l \ge 6\) for all l and introducing a 1/2 to the right-hand side of (9).

The corrected paper can be found at [2]. The author thanks A. Skopenkov for his interest in the paper and is grateful to him for pointing out the mistakes in it. See also [3].