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Statement

Setting (pp. 2, 4). $h_{a,d}(n)$ is the largest $t$ such that every $n$ points of $\mathbb R^d$ contain $t$ points whose non-zero volumes of $a$-element subsets are all distinct, and $H_{a,d}(t)$ is the least $n$ such that every $n$ points of $\mathbb R^d$ contain such $t$ points. $g_k(m,t)$ is as in Lemma 2.1.

Proposition 3.3 (p. 5, quoted). "For all integers $d\ge2$ and $t\ge d+1$, $H_{d+1,d}(t)\le g_{d+1}(2t,t)\le8t^{2d+2}$. In particular, $h_{d+1,d}(n)\ge n^{\frac{1}{2d+2}}/2$."

The introduction announces this as $h_{d+1,d}(n)\ge c_dn^{1/(2d+2)}$ (p. 2). It improves the exponent $1/((2d+1)d)$ that Theorem 1.2 gives at $a=d+1$. The paper adds (§5.1, p. 8) that for sets with no $d+1$ points on a hyperplane, counting all volumes, an almost identical proof gives $h'_{d+1,d}(n)\ge c_dn^{1/(2d+1)}$.

Proof pointer

P. 5. For a $d$-subset $D$ and a volume $\ell>0$, the points completing $D$ to volume $\ell$ form two hyperplanes parallel to that of $D$. If one holds $t$ of the points, those $t$ points have only zero volumes and suffice; otherwise the coloring of $(d+1)$-sets by volume, with zero-volume sets colored uniquely, is $2t$-good, and Lemma 2.1 with $k=d+1$, $m=2t$ gives $g_{d+1}(2t,t)\le8t^{2d+2}$.

Read depth

Claims checked: the statement and definitions were read clause by clause on the page images of arXiv:1401.6734v3. The proof was read for structure only, and nothing here is independently reviewed.

Dependencies

Lemma 2.1.

Source. D. Conlon, J. Fox, W. Gasarch, D. G. Harris, D. Ulrich and S. Zbarsky, Distinct volume subsets, SIAM J. Discrete Math. 29 (2015), 472--480, doi:10.1137/140954519; pages cited are those of the arXiv version arXiv:1401.6734v3, the edition named on the source card.

Bears on

None directly: the result concerns volumes of $(d+1)$-point simplices, not distances, and Problem 1208 asks about distances.