The unresolved measure-theoretic questions from Erdős's 1953 paper 'Some remarks on set theory. IV': Theorem 5 for mixed families of null and first-category sets, and positive-measure families with empty $\aleph_1$-subfamily intersections

Paul Erdős's 1953 paper 'Some remarks on set theory. IV' left open six questions: four geometric questions on decompositions of Euclidean spaces dual to partitions of their families of lines, and two measure-theoretic questions. The geometric questions were settled by a lineage running from Sierpiński's 1919 continuum-hypothesis-equivalent decomposition theorem through Davies (1963), Erdős–Jackson–Mauldin (1994, 1997), and Schmerl (1999): each finite-color variant is equivalent to the Continuum Hypothesis (more generally, to $2^{\aleph_0}\leq\aleph_m$), the one-point version of the countable-color question fails in ZFC, and the optimal ZFC bound for countable partitions of the lines of $\mathbb{R}^n$ is exactly two points. No corresponding settlement of the two measure-theoretic questions is known, and these form the open core recorded here. First, Erdős proved under the Continuum Hypothesis that a family of $\aleph_1$ sets of reals which does not cover $\mathbb{R}$ by countably many translates admits a set $A$ of cardinality $\aleph_1$ with $A+S_\alpha\neq\mathbb{R}$ for every family member $S_\alpha$; he remarked that he did not know whether the conclusion persists for a family consisting of $\aleph_1$ Lebesgue-null sets together with $\aleph_1$ first-category sets, the case in which the hypothesis can fail. Second, he asked whether there is a family of $2^{\aleph_0}$ positive-measure subsets of $\mathbb{R}$ of which every $\aleph_1$-sized subfamily has empty intersection; this is easy under the Continuum Hypothesis; on the other side, Martin's Axiom for $\aleph_1$ rules such a family out (an argument recorded as Exercise 27 of Chapter II of Kunen 1980), while no characterization between these two hypotheses is known. Both questions remain open: (1) as stated, and (2) as the problem of characterizing when such a family exists.

Background

Two classical smallness notions for subsets of the real line $\mathbb{R}$ run in parallel. A set is Lebesgue null (of measure zero) if, for every $\varepsilon>0$, it can be covered by countably many intervals of total length at most $\varepsilon$; a set is of first category (meager) if it is a countable union of nowhere dense sets, that is, of sets whose closure has empty interior. Both classes are $\sigma$-ideals — closed under taking subsets and under countable unions — and both are preserved by translations. They are nevertheless far from identical: there exist comeager sets of measure zero, and it follows that $\mathbb{R}$ is the union of a single null set and a single meager set, even though no countable union of translates of null sets covers $\mathbb{R}$, and likewise for meager sets. Throughout, $\aleph_1$ denotes the first uncountable cardinal, $2^{\aleph_0}$ the cardinality of the continuum, and the Continuum Hypothesis (CH) the assertion $2^{\aleph_0}=\aleph_1$. For $A,S\subseteq\mathbb{R}$, the algebraic sum $A+S=\{a+s:a\in A,s\in S\}$ is the union of the translates $a+S$ over $a\in A$.

The 1953 paper of Erdős sits at the border of cardinal arithmetic, measure, and category and asks two kinds of questions. Its geometric questions concern decompositions dual to partitions of lines: if the set of lines of $\mathbb{R}^n$ is partitioned into classes $L_i$, is there a partition of the point set, $\mathbb{R}^n=\bigcup_iS_i$, such that each line $\ell\in L_i$ meets $S_i$ in a small (finite, or at most $m$-point) set? The ancestry is a 1919 theorem of Sierpiński, equivalent to CH, of this type for fixed coordinate directions; Davies (1963) proved that CH is equivalent to the planar version with arbitrary directions, and in its strongest form (recorded as Theorem 0.1 of Schmerl's 1999 paper) for every finite $m$ the statement $2^{\aleph_0}\leq\aleph_m$ is equivalent to the assertion that every partition of the lines of $\mathbb{R}^n$ ($n\geq 2$) into $m+2$ classes admits such a point partition with finite intersections. Erdős–Jackson–Mauldin (1994) derived a sharp ordinal-indexed generalization and answered the 1953 three-class questions: the answer is yes exactly under CH. The countable-class variant took longer: Erdős asked whether countably many line classes always admit a point partition meeting each line of its class in at most one point; Davies (1963) refuted this, Erdős–Jackson–Mauldin (1994) refuted it in ZFC via a partition theorem of Todorčević, and Schmerl (1999) proved that two points always suffice in ZFC, with two optimal. The four geometric questions of the 1953 paper are thus completely settled.

The present record concerns the two measure-theoretic questions of the same paper, which are untouched by that lineage. The first concerns the paper's Theorem 5, an additive-avoidance statement: under CH, if $\{S_\alpha:\alpha<\omega_1\}$ is a family of subsets of $\mathbb{R}$ such that $\mathbb{R}$ is not a countable union of translates of members of the family, then there exists $A\subseteq\mathbb{R}$ with $|A|=\aleph_1$ such that $A+S_\alpha\neq\mathbb{R}$ for every $\alpha$. The hypothesis holds when all $S_\alpha$ are null (countably many translates of null sets are null) and when all $S_\alpha$ are meager (by the Baire category theorem, a countable union of meager sets cannot be $\mathbb{R}$); but it can fail for a mixed family, since a null set and a meager set can jointly cover $\mathbb{R}$. Erdős remarked that he did not know whether the conclusion nevertheless holds when the family consists of $\aleph_1$ null sets together with $\aleph_1$ first-category sets.

The second question asks whether there exists a family of $2^{\aleph_0}$ subsets of $\mathbb{R}$, each of positive Lebesgue measure, such that every $\aleph_1$-sized subfamily has empty intersection — equivalently (a reformulation), a point-countable family of positive-measure sets of size $2^{\aleph_0}$, where a family is point-countable if each real number belongs to at most countably many of its members. Erdős noted that such a family is easy to construct under CH; on the other side, a classical argument attributed to Exercise 27 of Chapter II of Kunen's 1980 book and cited on MathOverflow shows that Martin's Axiom for $\aleph_1$ blocks it. Martin's Axiom (MA) is a forcing axiom consistent with $2^{\aleph_0}>\aleph_1$ (forcing being the standard technique for building models of set theory with controlled continuum size); its instance MA($\aleph_1$) implies, among much else, that no union of $\aleph_1$ meager sets covers $\mathbb{R}$. What happens between CH and MA($\aleph_1$) — for instance, in models with $2^{\aleph_0}=\aleph_2$ in which MA($\aleph_1$) fails, or in terms of the cardinal characteristics of the continuum, the standard cardinal invariants (such as the additivity of the null ideal, the least number of null sets whose union has positive measure) that calibrate measure and category phenomena when CH fails — is not known.

Problem Statement

The 1953 paper left six questions; the four geometric ones are settled, and the surviving open core is the following two measure-theoretic questions. Their common objective is to complete the paper's analysis of uncountable families of small (null, first-category) and large (positive-measure) subsets of $\mathbb{R}$.

(1) The mixed-family extension of Erdős's additive-avoidance theorem. Theorem 5 of the 1953 paper, proved under CH, states that if $\{S_\alpha:\alpha<\omega_1\}$ is a family of subsets of $\mathbb{R}$ such that $\mathbb{R}$ is not a countable union of translates of members of the family, then there exists a set $A\subseteq\mathbb{R}$ of cardinality $\aleph_1$ with $A+S_\alpha\neq\mathbb{R}$ for every $\alpha$. That hypothesis is automatically satisfied when all members of the family are Lebesgue null, or when all are of first category, but it can fail for a mixed family, since a null set and a first-category set can jointly cover $\mathbb{R}$. The question, to be answered with or without CH: let $\{S_\alpha:\alpha<\omega_1\}$ be a family of subsets of $\mathbb{R}$ that is the union of $\aleph_1$ Lebesgue-null sets and $\aleph_1$ first-category sets (so the family has $\aleph_1$ members in all). Must there exist a set $A\subseteq\mathbb{R}$ of cardinality $\aleph_1$ such that $A+S_\alpha\neq\mathbb{R}$ for every $\alpha$? A negative answer would consist of such a mixed family with $A+S_\alpha=\mathbb{R}$ for some $\alpha$ whenever $|A|=\aleph_1$.

(2) Erdős's point-countable family of positive-measure sets. Does there exist a family $\{P_\xi:\xi<2^{\aleph_0}\}$ of subsets of $\mathbb{R}$, each of positive Lebesgue measure, such that every subfamily of cardinality $\aleph_1$ has empty intersection (equivalently, every real belongs to at most countably many members)? The construction is easy under CH; on the other side, MA($\aleph_1$) rules such a family out (Exercise 27 of Chapter II of Kunen's 1980 book, cited on MathOverflow). The existence question is thus not decided by ZFC alone, and the open problem in (2) is the characterization of when such a family exists.

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