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Statement

The definitions are those of §7.1 (p. 46). An isosceles set is a set of points among any three of which at most two distances occur, so that every triangle it spans is isosceles. Throughout Chapter 7, $X=\{x_1,\ldots,x_v\}$ is an isosceles set in $\mathbb{R}^d$ whose affine hull is $\mathbb{R}^d$, and $\dim(X_1)$ is the dimension of the affine hull of $X_1\subset X$. The set $X$ is decomposable if it has a partition $X=X_1\cup X_2$ with $\operatorname{card}(X_2)>1$ and $X_1\ne\emptyset$ such that each point of $X_1$ is equidistant from all points of $X_2$, the common distance allowed to depend on the point of $X_1$; such a pair $(X_1,X_2)$ is a decomposition.

Theorem 7.2.2 (p. 47). If an isosceles set $X$ is indecomposable, then it is a two-distance set.

The preceding Lemma 7.2.1 (p. 46) records the geometry of a decomposition: if $(X_1,X_2)$ is a decomposition of $X$, then $\dim(X_1)+\dim(X_2)\le\dim(X)$; its proof shows that the affine hulls of $X_1$ and $X_2$ are orthogonal. The chapter's introduction (p. 46) states the chapter's aim as showing that isosceles sets can be decomposed into a collection of mutually "orthogonal" two-distance sets.

Source. A. Blokhuis, Few-distance sets, CWI Tract 7, Centrum voor Wiskunde en Informatica, Amsterdam, 1984; definitions §7.1 and Lemma 7.2.1 on printed p. 46, Theorem 7.2.2, Lemma 7.2.3 and Lemma 7.2.4 on p. 47, the proof of Lemma 7.2.4 on pp. 47--48. The edition read is identified in the source digest.

Read depth. Claims checked: the definitions, Lemma 7.2.1 and Theorem 7.2.2 were read clause by clause on the page images, and the proofs of Lemmas 7.2.3 and 7.2.4 were read in full. Nothing here is independently reviewed.

Proof pointer

Color each pair of distinct points of $X$ by the distance between them. Lemma 7.2.3 (p. 47): if $X$ is indecomposable, every color class, as a graph on all of $X$, is connected; for a disconnected class, a component $X_2$ with more than one point has every outside point joined to it in a single color, by the isosceles property, so $(X\setminus X_2,X_2)$ would be a decomposition. The coloring satisfies the hypotheses of Lemma 7.2.4, which then allows at most two colors, that is, at most two distances.

Dependencies

Lemmas 7.2.3 and 7.2.4 (p. 47), both within the tract.

Bears on

• Problem 503: the decomposition step of the proof of Theorem 7.2.5, the tract's upper bound on the size of an isosceles set in $\mathbb{R}^d$.