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:
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].
References
Parsa, S.: On the links of vertices in simplicial \(d\)-complexes embeddable in the Euclidean \(2d\)-space. Discrete Comput. Geom. 59(3), 663–679 (2018)
Parsa, S.: On the links of vertices in simplicial \(d\)-complexes embeddable in the Euclidean \(2d\)-space (2020). arXiv:1512.05164
Skopenkov, A.: A short exposition of Salman Parsa’s theorems on intrinsic linking and non-realizability. Discrete Comput. Geom. (2019). https://doi.org/10.1007/s00454-019-00158-y, extended version: arXiv:1808.08363
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The original article can be found online at https://doi.org/10.1007/s00454-017-9936-1.
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Parsa, S. Correction to: On the Links of Vertices in Simplicial d-Complexes Embeddable in the Euclidean 2d-Space. Discrete Comput Geom 64, 227–228 (2020). https://doi.org/10.1007/s00454-020-00191-2
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DOI: https://doi.org/10.1007/s00454-020-00191-2