Ardal 2011 chaotic orderings rationals reals

../

remark_1: Ardal, Brown and Jungić's remark that replacing 2 by k in the doubling construction and repeating Sections 2 to 4 gives a linear ordering of the reals with monotonic k-term arithmetic progressions but no monotonic (k+1)-term one, asserted without a written proof.

theorem_2_2: Ardal, Brown and Jungić's explicit linear ordering of the integers, the union of nested doubling orderings of the intervals from -2^(n-1) to 2^(n-1)-1, in which no integer lies between two others whose average it is.

theorem_3_1: Ardal, Brown and Jungić's linear ordering of the rationals with no monotonic three-term arithmetic progression, obtained by König's infinity lemma from chaotic orderings of finite sets of rationals.

theorem_4_1: Ardal, Brown and Jungić's linear ordering of the reals, comparing two reals through a chaotic ordering of the rationals at the first Hamel-basis coordinate where they differ, has no monotonic three-term arithmetic progression.

---

Hayri Ardal, Tom Brown and Veselin Jungić, Chaotic orderings of the rationals and reals, Amer. Math. Monthly 118 (2011), no. 10, 921--925; DOI 10.4169/amer.math.monthly.118.10.921.

The copy read for this card is an author copy (pdfTeX, compiled in November 2013) of five pages with a text layer, headed "Citation data: Hayri Ardal, Tom Brown, and Veselin Jungić, Chaotic orderings of the rationals and reals, Amer. Math. Monthly 118 (2011), 921–925". The published version was not compared, so the page and label references below are the author copy's (pp. 1--5). Provenance: the copy was obtained in the survey download set of September 2026; the download URL was not recorded; 88,819 bytes. Read status: claims checked; every statement below was read from the text layer. That author copy prints no copyright or license line, and its download URL was not recorded, so no host's terms could be checked; the term is unstated.

Contents

A linear ordering $\prec$ of a set $X\subseteq\mathbb R$ is chaotic when no element of $X$ that is the average of two other elements of $X$ lies between them in $\prec$; a monotonic $k$-term arithmetic progression is a set $\{a_0+id:0\le i\le k-1\}$, $d\ne0$, with $a_0\prec a_1\prec\cdots\prec a_{k-1}$ (p. 1). The introduction records that chaotic orderings of $\{1,\dots,n\}$ exist for every $n$ (Monthly problem E 2440, 1975), that every listing of the positive integers as a sequence, one-way or two-way infinite, contains a monotonic 3-term progression (Davis, Entringer, Graham and Simmons, the paper's [2]), that whether every one-way infinite listing must contain a monotonic 4-term progression is open, and that 5-term progressions can be avoided.

• Definition 2.1, Lemma 2.1 and Theorem 2.2 (p. 2): $A_1=\langle0,-1\rangle$ and $A_{n+1}=(2A_n)(2A_n+1)$ define chaotic orderings $A_n$ of $[-2^{n-1},2^{n-1}-1]$ (so $A_2=\langle0,-2,1,-1\rangle$), each extending the previous one (the outer terms of a progression have the same parity, so they lie in the same half, which is chaotic by induction); their union $<_{\mathbb Z}$ is a chaotic linear ordering of $\mathbb Z$, with $0$ smallest and $-1$ largest.
• Theorem 3.1 (p. 3; Section 3, pp. 2--3): $\mathbb Q$ has a chaotic linear ordering $<_{\mathbb Q}$, obtained by König's infinity lemma from the tree of chaotic orderings of the initial segments $\{r_1,\dots,r_n\}$ of an enumeration of $\mathbb Q$; each finite set of rationals has one, by scaling it into $\mathbb Z$ and restricting $<_{\mathbb Z}$.
• Theorem 4.1 (p. 4; Section 4, pp. 3--4): $\mathbb R$ has a chaotic linear ordering $<_{\mathbb R}$. With $B$ a basis of $\mathbb R$ over $\mathbb Q$ ordered by the usual order, two reals are compared by $<_{\mathbb Q}$ at the first basis coordinate where they differ; a progression $a+c=2b$ has $a_i+c_i=2b_i$ in every coordinate, so a monotonic one for $<_{\mathbb R}$ would give a monotonic one for $<_{\mathbb Q}$. The proof uses the axiom of choice in the form that every vector space has a basis; the note asks whether a choice-free construction exists and observes that $\mathbb R$ may be replaced by any field of characteristic $0$.
• Remark 1 (p. 4): replacing $2$ by $k\ge2$ in Definition 2.1 ($C_1=\langle0,-1,-2,\dots,-(k-1)\rangle$ and $C_{n+1}=(kC_n)(kC_n+1)\cdots(kC_n+k-1)$, so that $C_n$ is an ordering of $[-(k-1)k^{n-1},k^{n-1}-1]$) and repeating the arguments of Sections 2--4 gives a linear ordering of $\mathbb R$ with monotonic $k$-term progressions but no monotonic $(k+1)$-term progression. Stated without written proof.
• Remarks 2--4 (pp. 4--5): on the nonnegative integers, $a<_{\mathbb Z}b$ exactly when the binary digit sequence of $a$, least significant digit first, precedes that of $b$ lexicographically, that is when $\sum a_i2^{-i}<\sum b_i2^{-i}$; the same digit-reversal rule defines an explicit chaotic ordering $<_D$ of the nonnegative dyadic rationals, order-isomorphic to their usual order.

Compiled scope

Every statement above was read from the text layer of the five pages. The proofs of Lemma 2.1 and of Theorems 2.2, 3.1 and 4.1 were read in full and followed; Remark 1 is asserted by the authors without a written proof and was not checked. No proof is rewritten here and nothing has been independently reviewed.

Bears on. #194: Theorem 4.1 gives a linear ordering of $\mathbb R$ with no monotonic 3-term arithmetic progression, hence none of any length $k\ge3$, which answers the problem's question no for every $k\ge3$; it is built from Theorem 2.2 and Theorem 3.1. Remark 1 adds, without written proof, that for each $k\ge2$ some ordering of $\mathbb R$ has monotonic $k$-term progressions but no monotonic $(k+1)$-term one. The problem's claim page for this paper records the claim and its evidence; the formalization it lists is not examined here.

Results. Page numbers are those of the author copy read (pp. 1--5).

• Theorem 2.2 (p. 2), with Definition 2.1, Lemma 2.1 and Definition 2.2 (p. 2): the explicit chaotic ordering $<_{\mathbb Z}$ of $\mathbb Z$.
• Theorem 3.1 (p. 3; Section 3, pp. 2--3): a chaotic ordering $<_{\mathbb Q}$ of $\mathbb Q$.
• Theorem 4.1 (p. 4; Section 4, pp. 3--4): a chaotic ordering $<_{\mathbb R}$ of $\mathbb R$.
• Remark 1 (p. 4): the $k$-fold generalization, asserted without written proof.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.