 12 solved out of 48 shown
Let $k\geq 2$. Is there an integer $n_k$ such that, if $D=\{ 1<d<n_k : d\mid n_k\}$, then for any $k$-colouring of $D$ there is a monochromatic subset $D'\subseteq D$ such that $\sum_{d\in D'}\frac{1}{d}=1$?
This follows from the colouring result of Croot [Cr03]. Obtaining better quantitative aspects seems interesting still -- note that Croot's result allows for $n_k \leq e^{C^k}$ for some constant $C>1$. Is there a better bound?
Does every finite colouring of the integers have a monochromatic solution to $1=\sum \frac{1}{n_i}$ with $2\leq n_1<\cdots <n_k$?
The answer is yes, as proved by Croot [Cr03].
$100 If$\delta>0$and$N$is sufficiently large in terms of$\delta$, and$A\subseteq\{1,\ldots,N\}$is such that$\sum_{a\in A}\frac{1}{a}>\delta \log N$then must there exist$S\subseteq A$such that$\sum_{n\in S}\frac{1}{n}=1$? Solved by Bloom [Bl21], who showed that the quantitative threshold $\sum_{n\in A}\frac{1}{n}\gg \frac{\log\log\log N}{\log\log N}\log N$ is sufficient. Erdős further speculates that perhaps even$\gg (\log\log N)^2$might be sufficient. (A construction of Pomerance, as discussed in the appendix of [Bl21], shows that this would be best possible.) Let$F(k)$be the number of solutions to $1= \frac{1}{n_1}+\cdots+\frac{1}{n_k},$ where$1\leq n_1<\cdots<n_k$are distinct integers. Find good estimates for$F(k)$. The current best bounds known are $2^{c^{\frac{k}{\log k}}}\leq F(k) \leq c_0^{(\frac{1}{5}+o(1))2^k},$ where$c>0$is some absolute constant and$c_0=1.26408\cdots$is the 'Vardi constant'. The lower bound is due to Konyagin [Ko14] and the upper bound to Elsholtz and Planitzer [ElPl21]. Let$x\in \overline{\mathbb{Q}}$be an algebraic number. For any$n\geq 1$let $R_n(x) = \sum_{i=1}^n\frac{1}{m_i}<x$ be the maximal sum of$n$distinct unit fractions which is$<x$. Is it true that, for sufficiently large$n$, we have $R_{n+1}(x)=R_n(x)+\frac{1}{m},$ where$m$is minimal such that$m$does not appear in$R_n(x)$and the right-hand side is$<x$? (That is, are the best underapproximations eventually always constructed in a 'greedy' fashion?) If$x$is irrational then this can be false, but it may be true for almost all$x\in\mathbb{R}$. Curtiss [Cu22] showed that this is true for$x=1$and Erdős [Er50b] showed it is true for all$x\in\mathbb{Q}$. For every$n\geq 2$there exist distinct integers$1\leq x<y<z$such that $\frac{4}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ The Erdős-Straus conjecture. The existence of a representation of$4/n$as the sum of at most four distinct unit fractions follows trivially from a greedy algorithm. Schinzel conjectured the generalisation that, for any fixed$a$, if$n$is sufficiently large in terms of$a$then there exist distinct integers$1\leq x<y<z$such that $\frac{a}{n} = \frac{1}{x}+\frac{1}{y}+\frac{1}{z}.$ Let$A\subseteq \mathbb{N}$be an infinite set and consider the following greedy algorithm for a rational$x\in (0,1)$: choose the minimal$n\in A$such that$n\geq 1/x$and repeat with$x$replaced by$x-\frac{1}{n}$. If this terminates after finitely many steps then this produces a representation of$x$as the sum of distinct unit fractions with denominators from$A$. Does this process always terminate if$x$has odd denominator and$A$is the set of odd numbers? More generally, for which pairs$x$and$A$does this process terminate? In 1202 Fibonacci observed that this process terminates for any$x$when$A=\mathbb{N}$. The problem when$A$is the set of odd numbers is due to Stein. Graham [Gr64b] has shown that$\frac{m}{n}$is the sum of distinct unit fractions with denominators$\equiv a\pmod{d}$if and only if $\left(\frac{n}{(n,(a,d))},\frac{d}{(a,d)}\right)=1.$ Does the greedy algorithm always terminate in such cases? Graham [Gr64c] has also shown that$x$is the sum of distinct unit fractions with square denominators if and only if$x\in [0,\pi^2/6-1)\cup [1,\pi^2/6)$. Does the greedy algorithm for this always terminate? Erdős and Graham believe not - indeed, perhaps it fails to terminate almost always. Additional thanks to: Zachary Hunter Let$p:\mathbb{Z}\to \mathbb{Z}$be a polynomial whose leading coefficient is positive and such that there exists no$d\geq 2$with$d\mid p(n)$for all$n\geq 1$. Is it true that, for all sufficiently large$m$, there exist integers$1\leq n_1<\cdots <n_k$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ and $m=p(n_1)+\cdots+p(n_k)?$ Graham [Gr63] has proved this when$p(x)=x$. Cassels [Ca60] has proved that these conditions on the polynomial imply every sufficiently large integer is the sum of$p(n_i)$with distinct$n_i$. Burr has proved this if$p(x)=x^k$with$k\geq 1$and if we allow$n_i=n_j$. Let$f(k)$be the maximal value of$n_1$such that there exist$n_1<n_2<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $f(k)=(1+o(1))\frac{k}{e-1}?$ The upper bound$f(k) \leq (1+o(1))\frac{k}{e-1}$is trivial since for any$u\geq 1$we have $\sum_{u\leq n\leq eu}\frac{1}{n}=1+o(1),$ and hence if$f(k)=u$then we must have$k\geq (e-1-o(1))u$. Essentially solved by Croot [Cr01], who showed that for any$N>1$there exists some$k\geq 1$and $N<n_1<\cdots <n_k \leq (e+o(1))N$ with$1=\sum \frac{1}{n_i}$. Additional thanks to: Zachary Hunter Let$f(k)$be the minimal value of$n_k$such that there exist$n_1<n_2<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $f(k)=(1+o(1))\frac{e}{e-1}k?$ It is trivial that$f(k)\geq (1+o(1))\frac{e}{e-1}k$, since for any$u\geq 1$$\sum_{e\leq n\leq eu}\frac{1}{n}= 1+o(1),$ and so if$eu\approx f(k)$then$k\leq \frac{e-1}{e}f(k)$. Proved by Martin [Ma00]. Additional thanks to: Zachary Hunter Let$k\geq 2$. Is it true that there exists an interval$I$of width$(e-1+o(1))k$and integers$n_1<\cdots<n_k\in I$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?$ The answer is yes, proved by Croot [Cr01]. Let$k\geq 2$. Is it true that, for any distinct integers$n_1<\cdots <n_k$such that $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ we must have$\max(n_{i+1}-n_i)\geq 3$? The example$1=\frac{1}{2}+\frac{1}{3}+\frac{1}{6}$shows that$3$would be best possible here. The lower bound of$\geq 2$is equivalent to saying that$1$is not the sum of reciprocals of consecutive integers, proved by Erdős [Er32]. This conjecture would follow for all but at most finitely many exceptions if it were known that, for all large$N$, there exists a prime$p\in [N,2N]$such that$\frac{p+1}{2}$is also prime. Is it true that there are only finitely many pairs of intervals$I_1,I_2$such that $\sum_{n_1\in I_1}\frac{1}{n_1}+\sum_{n_2\in I_2}\frac{1}{n_2}\in \mathbb{N}?$ This is still open even if$\lvert I_2\rvert=1$. It is perhaps true with two intervals replaced by any$k$intervals. Is it true that, for all sufficiently large$k$, there exist intervals$I_1,\ldots,I_k$with$\lvert I_i\rvert \geq 2$for$1\leq i\leq k$such that $1=\sum_{i=1}^k \sum_{n\in I_i}\frac{1}{n}?$ Let$a\geq 1$. Must there exist some$b>a$such that $\sum_{a\leq n\leq b}\frac{1}{n}=\frac{r_1}{s_1}\textrm{ and }\sum_{a\leq n\leq b+1}\frac{1}{n}=\frac{r_2}{s_2},$ with$(r_i,s_i)=1$and$s_2<s_1$? If so, how does this$b(a)$grow with$a$? For example, $\sum_{3\leq n\leq 5}\frac{1}{n} = \frac{47}{60}\textrm{ and }\sum_{3\leq n\leq 6}\frac{1}{n}=\frac{19}{20}.$ Let$n\geq 1$and define$L_n$to be the lowest common multiple of$\{1,\ldots,n\}$and$a_n$by $\sum_{1\leq k\leq n}\frac{1}{k}=\frac{a_n}{L_n}.$ Is it true that$(a_n,L_n)=1$and$(a_n,L_n)>1$both occur for infinitely many$n$? Let$A$be the set of$n\in \mathbb{N}$such that there exist$1\leq m_1<\cdots <m_k=n$with$\sum\tfrac{1}{m_i}=1$. Explore$A$. In particular, • Does$A$have density$1$? • What are those$n\in A$not divisible by any$d\in A$with$1<d<n$? Straus observed that$A$is closed under multiplication. Furthermore, it is easy to see that$A$does not contain any prime power. Additional thanks to: Zachary Hunter Let$k\geq 1$and let$v(k)$be the minimal integer which does not appear as some$n_i$in a solution to $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ with$1\leq n_1<\cdots <n_k$. Estimate the growth of$v(k)$. Results of Bleicher and Erdős [BlEr75] imply$v(k) \gg k!$. It may be that$v(k)$grows doubly exponentially in$\sqrt{k}$or even$k$. An elementary inductive argument shows that$n_k\leq ku_k$where$u_1=1$and$u_{i+1}=u_i(u_i+1)$, and hence $v(k) \leq kc_0^{2^k},$ where $c_0=\lim_n u_n^{1/2^n}=1.26408\cdots$ is the 'Vardi constant'. Additional thanks to: Zachary Hunter Let$N\geq 1$and let$t(N)$be the least integer$t$such that there is no solution to $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}$ with$t=n_1<\cdots <n_k\leq N$. Estimate$t(N)$. Erdős and Graham [ErGr80] claim a bound of $t(N)\ll\frac{N}{\log N},$ but have no idea of the true value of$t(N)$. Let$N\geq 1$and let$k(N)$denote the smallest$k$such that there exist$N\leq n_1<\cdots <n_k$with $1=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Is it true that $\lim_{N\to \infty} k(N)-(e-1)N=\infty?$ Erdős and Straus [ErSt71b] have proved the existence of some constant$c>0$such that $-c < k(N)-(e-1)N \ll \frac{N}{\log N}.$ Let$N\geq 1$and let$k(N)$be maximal such that there are$k$disjoint$A_1,\ldots,A_k\subseteq \{1,\ldots,N\}$with$\sum_{n\in A_i}\frac{1}{n}=1$for all$i$. Estimate$k(N)$. Is it true that$k(N)=o(\log N)$? More generally, how many disjoint$A_i$can be found in$\{1,\ldots,N\}$such that the sums$\sum_{n\in A_i}\frac{1}{n}$are all equal? Using sunflowers it can be shown that there are at least$N\exp(-O(\sqrt{\log N}))$such sets. Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl23] (coupled with the trivial greedy approach) implies that$k(N)=(1-o(1))\log N$. Additional thanks to: Zachary Hunter and Mehtaab Sawhney Let$N\geq 1$. How many$A\subseteq \{1,\ldots,N\}$are there such that$\sum_{n\in A}\frac{1}{n}=1$? It is not even known whether this is$2^{cN}$for some$c<1$or$2^{(1+o(1))N}$. Does every set$A\subseteq \mathbb{N}$of positive density contain some finite$S\subset A$such that$\sum_{n\in S}\frac{1}{n}=1$? The answer is yes, proved by Bloom [Bl21]. Is there an infinite sequence$a_1<a_2<\cdots $such that$a_{i+1}-a_i=O(1)$and no finite sum of$\frac{1}{a_i}$is equal to$1$? There does not exist such a sequence, which follows from the positive solution to  by Bloom [Bl21]. Let$A(N)$denote the maximal cardinality of$A\subseteq \{1,\ldots,N\}$such that$\sum_{n\in S}\frac{1}{n}\neq 1$for all$S\subseteq A$. Estimate$A(N)$. Erdős and Graham believe the answer is$A(N)=(1+o(1))N$. Croot [Cr03] disproved this, showing the existence of some constant$c<1$such that$A(N)<cN$for all large$N$. It is trivial that$A(N)\geq (1-\frac{1}{e}+o(1))N$. Additional thanks to: Zachary Hunter What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that for any$a,b_1,\ldots,b_k\in A$with$k\geq 2$we have $\frac{1}{a}\neq \frac{1}{b_1}+\cdots+\frac{1}{b_k}?$ Is there a constant$c>1/2$such that$\lvert A\rvert >cN$is possible for all large$N$? The example$A=(N/2,N]\cap \mathbb{N}$shows that$\lvert A\rvert\geq N/2$. Additional thanks to: Zachary Hunter Is it true that, for any$\delta>1/2$, if$N$is sufficiently large and$A\subseteq \{1,\ldots,N\}$has$\lvert A\rvert \geq \delta N$then there exist$a,b,c\in A$such that $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}.$ The colouring version of this is , which was solved by Brown and Rödl [BrRo91]. The possible alternative question, that if$A\subseteq \mathbb{N}$is a set of positive lower density then must there exist$a,b,c\in A$such that $\frac{1}{a}=\frac{1}{b}+\frac{1}{c},$ has a negative answer, taking for example$A$to be the union of$[5^k,(1+\epsilon)5^k]$for large$k$and sufficiently small$\epsilon>0$. This was observed by Hunter and Sawhney. Additional thanks to: Zachary Hunter and Mehtaab Sawhney Is it true that in any finite colouring of the integers there exists a monochromatic solution to $\frac{1}{a}=\frac{1}{b}+\frac{1}{c}$ with distinct$a,b,c$? The density version of this is . This colouring version is true, as proved by Brown and Rödl [BrRo91]. For integers$1\leq a<b$let$N(a,b)$denote the minimal$k$such that there exist integers$1<n_1<\cdots<n_k$with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Estimate$N(b)=\max_{1\leq a<b}N(a,b)$. Is it true that$N(b) \ll \log\log b$? Erdős [Er50c] proved that $\log\log b \ll N(b) \ll \frac{\log b}{\log\log b}.$ The upper bound was improved by Vose [Vo85] to $N(b) \ll \sqrt{\log b}.$ One can also investigate the average of$N(a,b)$for fixed$b$, and it is known that $\frac{1}{b}\sum_{1\leq a<b}N(a,b) \gg \log\log b.$ Related to . Additional thanks to: Zachary Hunter For integers$1\leq a<b$let$D(a,b)$be the minimal value of$n_k$such that there exist integers$1\leq n_1<\cdots <n_k$with $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}.$ Estimate$D(b)=\max_{1\leq a<b}D(a,b)$. Is it true that $D(b) \ll b(\log b)^{1+o(1)}?$ Bleicher and Erdős [BlEr76] have shown that $D(b)\ll b(\log b)^2.$ If$b=p$is a prime then $D(p) \gg p\log p.$ Let$a/b\in \mathbb{Q}_{>0}$with$b$squarefree. Are there integers$1<n_1<\cdots<n_k$, each the product of two distinct primes, such that $\frac{a}{b}=\frac{1}{n_1}+\cdots+\frac{1}{n_k}?$ For$n_i$the product of three distinct primes, this is true when$b=1$, as proved by Butler, Erdős and Graham [BuErGr15] (this paper is perhaps Erdős' last paper, appearing 19 years after his death). Are there two finite sets of primes$P,Q$such that $1=\left(\sum_{p\in P}\frac{1}{p}\right)\left(\sum_{q\in Q}\frac{1}{q}\right)?$ Asked by Barbeau [Ba76]. Can this be done if we drop the requirement that all$p\in P$are prime and just ask for them to be relatively coprime, and similarly for$Q$? Let$N\geq 1$. What is the smallest integer not representable as the sum of distinct unit fractions with denominators from$\{1,\ldots,N\}$? Is it true that the set of integers not representable as such has the shape$[m,\infty)$for some$m$? Let$N\geq 1$. How many integers can be written as the sum of distinct unit fractions with denominators from$\{1,\ldots,N\}$? Are there$o(\log N)$such integers? Hunter and Sawhney have observed that Theorem 3 of Bloom [Bl23] implies the answer to the second question is no: in fact all integers$m\leq (1-o(1))\log N$can be written as the sum of distinct unit fractions with denominators from$\{1,\ldots,N\}$. Additional thanks to: Zachary Hunter and Mehtaab Sawhney Let$\alpha >0$and$N\geq 1$. Is it true that, for any$A\subseteq \{1,\ldots,N\}$with$\lvert A\rvert \geq \alpha N$there exists some$S\subseteq A$such that $\frac{a}{b}=\sum_{n\in S}\frac{1}{n}$ with$b =O_\alpha(1)$? What is the minimal value of$\lvert 1-\sum_{n\in A}\frac{1}{n}\rvert$as$A$ranges over all subsets of$\{1,\ldots,N\}$which contain no$S$such that$\sum_{n\in S}\frac{1}{n}=1$? Is it $e^{-(c+o(1))N}$ for some constant$c\in (0,1)$? It is trivially at least$1/[1,\ldots,N]$. Does there exist some$c>0$such that, for any$K>1$, whenever$A$is a sufficiently large finite multiset of integers with$\sum_{n\in A}\frac{1}{n}>K$there exists some$S\subseteq A$such that $1-e^{-cK} < \sum_{n\in S}\frac{1}{n}\leq 1?$ Erdős and Graham knew this with$e^{-cK}$replaced by$c/K^2$. Additional thanks to: Mehtaab Sawhney Are there infinitely many solutions to $\frac{1}{p_1}+\cdots+\frac{1}{p_k}=1-\frac{1}{m},$ where$m\geq 2$is an integer and$p_1<\cdots<p_k$are distinct primes? For example, $\frac{1}{2}+\frac{1}{3}=1-\frac{1}{6}.$ Let$n\geq 1$and let$t$be minimal such that$\sum_{n\leq k\leq t}\frac{1}{k}\geq 1$. We define $\epsilon(n) = \sum_{n\leq k\leq t}\frac{1}{k}-1.$ How small can$\epsilon(n)$be? Is it true that $\liminf n^2\epsilon(n)=0?$ It is perhaps true that$n^{2+\delta}\epsilon(n)\to \infty$for all$\delta >0$. Let$u_1=1$and$u_{n+1}=u_n(u_n+1)$. Let$a_1<a_2<\cdots $be any other sequence with$\sum \frac{1}{a_k}=1$. Is it true that $\liminf a_n^{1/2^n}<\lim u_n^{1/2^n}=c_0=1.264085\cdots?$$c_0$is called the Vardi constant. Is it true that if$A\subset \mathbb{N}\backslash\{1\}$is a finite set with$\sum_{n\in A}\frac{1}{n}<2$then there is a partition$A=A_1\sqcup A_2$such that $\sum_{n\in A_i}\frac{1}{n}<1$ for$i=1,2$? This is not true if$A$is a multiset, for example$2,3,3,5,5,5,5$. This is not true in general, as shown by Sándor [Sa97], who observed that the proper divisors of$120$form a counterexample. More generally, Sándor shows that for any$n\geq 2$there exists a finite set$A\subseteq \mathbb{N}\backslash\{1\}$with$\sum_{k\in A}\frac{1}{k}<n$and no partition into$n$parts each of which has$\sum_{k\in A_i}\frac{1}{k}<1$. The minimal counterexample is$\{2,3,4,5,6,7,10,11,13,14,15\}$, found by Tom Stobart. Additional thanks to: Tom Stobart Is there some constant$c>0$such that for every$n\geq 1$there exists some$\delta_k\in \{-1,0,1\}$for$1\leq k\leq n$with $0< \left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert < \frac{c}{2^n}?$ Is it true that for sufficiently large$n$, for any$\delta_k\in \{-1,0,1\}$, $\left\lvert \sum_{1\leq k\leq n}\frac{\delta_k}{k}\right\rvert > \frac{1}{[1,\ldots,n]}$ whenever the left-hand side is not zero? Inequality is obvious for the second claim, the problem is strict inequality. This fails for small$n$, for example $\frac{1}{2}-\frac{1}{3}-\frac{1}{4}=-\frac{1}{12}.$ Additional thanks to: Zachary Chase Let$A\subseteq \mathbb{N}$be an infinite arithmetic progression and$f:A\to \{-1,1\}$be a non-constant function. Must there exist a finite$S\subset A$such that $\sum_{n\in S}\frac{f(n)}{n}=0?$ What about if$A$is an arbitrary set of positive density? What if$A$is the set of squares excluding$1$? Erdős and Straus [ErSt75] proved this when$A=\mathbb{N}$. Sattler [Sat75] proved this when$A$is the set of odd numbers. For the squares$1$must be excluded or the result is trivially false, since $\sum_{k\geq 2}\frac{1}{k^2}<2.$ What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that there is a function$\delta:A\to \{-1,1\}$such that $\sum_{n\in A}\frac{\delta_n}{n}=0$ and $\sum_{n\in A'}\frac{\delta_n}{n}\neq 0$ for all$A'\subsetneq A$? Let$S(N)$count the number of distinct sums of the form$\sum_{n\in A}\frac{1}{n}$for$A\subseteq \{1,\ldots,N\}$. Estimate$S(N)$. Bleicher and Erdős [BlEr75] proved the lower bound $\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq \frac{\log S(N)}{\log 2},$ valid for$k\geq 4$and$\log_kN\geq k$, and also [BlEr76b] proved the upper bound $\log S(N)\leq \log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),$ valid for$r\geq 1$and$\log_{2r}N\geq 1$. (In these bounds$\log_in$denotes the$i$-fold iterated logarithm.) See also . Additional thanks to: Boris Alexeev and Dustin Mixon What is the size of the largest$A\subseteq \{1,\ldots,N\}$such that all sums$\sum_{n\in S}\frac{1}{n}$are distinct for$S\subseteq A$? Let$R(N)$be the maximal such size. Results of Bleicher and Erdős from [BlEr75] and [BlEr76b] imply that $\frac{N}{\log N}\prod_{i=3}^k\log_iN\leq R(N)\leq \frac{1}{\log 2}\log_r N\left(\frac{N}{\log N} \prod_{i=3}^r \log_iN\right),$ valid for any$k\geq 4$with$\log_kN\geq k$and any$r\geq 1$with$\log_{2r}N\geq 1$. (In these bounds$\log_in$denotes the$i$-fold iterated logarithm.) See also . Additional thanks to: Boris Alexeev, Zachary Hunter, and Dustin Mixon Suppose$A\subseteq \{1,\ldots,N\}$is such that if$a,b\in A$then$a+b\nmid ab$. Can$A$be 'substantially more' than the odd numbers? What if$a,b\in A$with$a\neq b$implies$a+b\nmid 2ab$? Must$\lvert A\rvert=o(N)$? The connection to unit fractions comes from the observation that$\frac{1}{a}+\frac{1}{b}$is a unit fraction if and only if$a+b\mid ab$. Let$A\subseteq \mathbb{N}$be a lacunary sequence (so that$A=\{a_1<a_2<\cdots\}$and there exists some$\lambda>1$such that$a_{n+1}/a_n\geq \lambda$for all$n\geq 1\$). Must $\left\{ \sum_{a\in A'}\frac{1}{a} : A'\subseteq A\textrm{ finite}\right\}$ contain all rationals in some open interval?
Bleicher and Erdős conjecture the answer is no.