Ulam asked in 1945 if there is an everywhere dense \emph{rational set}, i.e. a point set in the plane with all its pairwise distances rational. Erd\H os conjectured that if a set $S$ has a dense rational subset, then $S$ should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erd\H os's conjecture for algebraic curves, by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.
Resolving Erdős-Ulam Monochromatic Union-Closed Family Conjectures
Electro-Gravitational Space Propulsion Laboratory (EGSPL)（电引力空间推进实验室）；Maulana Azad National Institute of Technology Bhopal（莫拉纳·阿扎德国家技术学院博帕尔分校）；National Taiwan University（国立台湾大学）
A rational distance set is a subset of the plane such that the distance between any two points is a rational number. We show, assuming Lang's Conjecture, that the cardinalities of rational distance sets in general position are uniformly bounded, generalizing results of Solymosi-de Zeeuw, Makhul-Shaffaf, Shaffaf, and Tao. In the process, we give a criterion for certain varieties with non-canonical singularities to be of general type.
The main aim of this paper is to bridge two directions of research generalizing asymptotic density zero sets. This enables to transfer results concerning one direction to the other one. Consider a function $g\colonω\to [0,\infty)$ such that $\lim_{n\to\infty}g(n)=\infty$ and $\frac{n}{g(n)}$ does not converge to $0$. Then the family $\mathcal{Z}_g=\{A\subseteqω:\ \lim_{n\to\infty}\frac{\text{card}(A\cap n)}{g(n)}=0\}$ is an ideal called simple density ideal (or ideal associated to upper density of weight $g$). We compare this class of ideals with Erdős-Ulam ideals. In particular, we show that …
We study the problem of identifying an initially unknown $m$-bit number by using yes-no questions when up to a fixed number $e$ of the answers can be erroneous. In the variant we consider here questions are restricted to be the union of up to a fixed number of intervals. For any $e \geq 1$ let $k_e$ be the minimum $k$ such that for all sufficiently large $m$, there exists a strategy matching the information theoretic lower bound and only using $k$-interval questions. It is known that $k_e = O(e^2)$. However, it has been conjectured that the $k_e = Θ(e).$ This linearity conjecture is supported …
Erdős & Graham ask whether the equation $x^2+y^2=z^2$ is partition regular, i.e. whether it has a finite Rado number. This note provides a lower bound and also states results in the affirmative for two similar quadratic equations.
There exists an infinite family of examples of subsets of $\mathbb{F}_q^2$ with $q^{4/3}$ elements whose distance sets are not the whole of $\mathbb{F}_q$.
A covering system is a finite collection of arithmetic progressions whose union is the set of integers. The study of these objects was initiated by Erdős in 1950, and over the following decades he asked many questions about them. Most famously, he asked whether there exist covering systems with distinct moduli whose minimum modulus is arbitrarily large. This problem was resolved in 2015 by Hough, who showed that in any such system the minimum modulus is at most $10^{16}$. The purpose of this note is to give a gentle exposition of a simpler and stronger variant of Hough's method, which was rece…
Introduced by Erdős in 1950, a covering system of the integers is a finite collection of arithmetic progressions whose union is the set $\mathbb{Z}$. Many beautiful questions and conjectures about covering systems have been posed over the past several decades, but until recently little was known about their properties. Most famously, the so-called minimum modulus problem of Erdős was resolved in 2015 by Hough, who proved that in every covering system with distinct moduli, the minimum modulus is at most $10^{16}$. In this paper we answer another question of Erdős, asked in 1952, on the number o…
An edge-coloring of a complete graph with a set of colors $C$ is called completely balanced if any vertex is incident to the same number of edges of each color from $C$. Erdős and Tuza asked in $1993$ whether for any graph $F$ on $\ell$ edges and any completely balanced coloring of any sufficiently large complete graph using $\ell$ colors contains a rainbow copy of $F$. This question was restated by Erdős in his list of ``Some of my favourite problems on cycles and colourings''. We answer this question in the negative for most cliques $F=K_q$ by giving explicit constructions of respective comp…
A set of integers greater than 1 is primitive if no element divides another. Erdős proved in 1935 that the sum of $1/(n \log n)$ for $n$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained by the set of prime numbers. We answer the Erdős question in the affirmative for 2-primitive sets. Here a set is 2-primitive if no element divides the product of 2 other elements.
Motivated by the Erdos-Szekeres convex polytope conjecture in $R^d$, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers $ n > k \geq 5$, what is the minimum integer $g_k(n)$ such that any $k$-uniform hypergraph on $g_k(n)$ vertices with the property that any set of $k + 1$ vertices induces 0, 2, or 4 edges, contains an independent set of size $n$. Our main result shows that $g_k(n) > 2^{cn^{k-4}}$, where $c = c(k)$.
For given simple graphs $G_1$ and $G_2$, the size Ramsey number $\hat{R}(G_1,G_2)$ is the smallest positive integer $m$, where there exists a graph $G$ with $m$ edges such that in any edge coloring of $G$ with two colors red and blue, there is either a red copy of $G_1$ or a blue copy of $G_2$. In 1981, Erdős and Faudree investigated the size Ramsey number $\hat{R}(K_n,tK_2)$, where $K_n$ is a complete graph on $n$ vertices and $tK_2$ is a matching of size $t$. They obtained the value of $\hat{R}(K_n,tK_2)$ when $n\geq 4t-1$ as well as for $t=2$ and asked for the behavior of these numbers when…
Since their introduction by Erdős in 1950, covering systems (that is, finite collections of arithmetic progressions that cover the integers) have been extensively studied, and numerous questions and conjectures have been posed regarding the existence of covering systems with various properties. In particular, Erdős asked if the moduli can be distinct and all arbitrarily large, Erdős and Selfridge asked if the moduli can be distinct and all odd, and Schinzel conjectured that in any covering system there exists a pair of moduli, one of which divides the other. Another beautiful conjecture, propo…
We develop novel techniques which allow us to prove a diverse range of results relating to subset sums and complete sequences of positive integers, including solutions to several longstanding open problems. These include: solutions to the three problems of Burr and Erdős on Ramsey complete sequences, for which Erdős later offered a combined total of \$350; analogous results for the new notion of density complete sequences; the solution to a conjecture of Alon and Erdős on the minimum number of colors needed to color the positive integers less than $n$ so that $n$ cannot be written as a monochr…
A family of sets is intersecting if no two of its members are disjoint, and has the Erdős-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection. Denote by $\mathcal{H}_k(n,p)$ the random family in which each $k$-subset of $\{1\dots n\}$ is present with probability $p$, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks: \[ \mbox{for what $p=p(n,k)$ is $\mathcal{H}_k(n,p)$ likely to be EKR?} \] Here, for fixed $c<1/4$, and $k< \sqrt{cn\log n}$ we give a precise answer to this question, characterizing those se…
More than 50 years ago, Erd\H os asked the following question: what is the maximum size of a family $\mathcal F$ of $k$-element subsets of an $n$-element set if it has no $s+1$ pairwise disjoint sets? This question attracted a lot of attention recently, in particular, due to its connection to various combinatorial, probabilistic and theoretical computer science problems. Improving the previous best bound due to the first author, we prove that $|\mathcal F|\le {n\choose k}-{n-s\choose k}$, provided $n\ge \frac 53sk -\frac 23 s$ and $s$ is sufficiently large. We derive several corollaries concer…
We prove that the set of long cycles has the edge-Erdős-Pósa property: for every fixed integer $\ell\ge 3$ and every $k\in\mathbb{N}$, every graph $G$ either contains $k$ edge-disjoint cycles of length at least $\ell$ (long cycles) or an edge set $X$ of size $O(k^2\log k + \ell k)$ such that $G-X$ does not contain any long cycle. This answers a question of Birmelé, Bondy, and Reed (Combinatorica 27 (2007), 135--145).
Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed $\varepsilon >0$ such that if $n=2k+1$ and $p> 1-\varepsilon$, then w.h.p. (that is, with probability tending to 1 as $k\rightarrow \infty$), $\mathcal{H}_k (n,p)$ has the "Erdős-Ko-Rado property." We also mention a similar random version of Sperner's Theorem.
This paper studies a problem of Erdös concerning lattice cubes. Given an $N \times N \times N$ lattice cube, we want to find the maximum number of vertices one can select so that no eight corners of a rectangular box are chosen simultaneously. Erdös conjectured that it has a sharp upper bound, which is $O(N^{11/4})$, but no example that large has been found yet. We start approaching this question for small $N$ using the method of exhaustion, and we find that there is not necessarily a unique maximal set of vertices (counting all possible symmetries). Next, we study an equivalent two-dimensiona…
Many recent results in combinatorics concern the relationship between the size of a set and the number of distances determined by pairs of points in the set. One extension of this question considers configurations within the set with a specified pattern of distances. In this paper, we use graph-theoretic methods to prove that a sufficiently large set $E$ must contain at least $C_G|E|$ distinct copies of any given weighted tree $G$, where $C_G$ is a constant depending only on the graph $G$.
Let $Δ_s=R(K_3,K_s)-R(K_3,K_{s-1})$, where $R(G,H)$ is the Ramsey number of graphs $G$ and $H$ defined as the smallest $n$ such that any edge coloring of $K_n$ with two colors contains $G$ in the first color or $H$ in the second color. In 1980, Erdős and Sós posed some questions about the growth of $Δ_s$. The best known concrete bounds on $Δ_s$ are $3 \le Δ_s \le s$, and they have not improved since the stating of the problem. In this paper we present some constructions, which imply in particular that $R(K_3,K_s) \ge R(K_3,K_{s-1}-e) + 4$. This does not improve the lower bound of 3 on $Δ_s$, b…
In this paper we prove several results on Ramsey numbers $R(H,F)$ for a fixed graph $H$ and a large graph $F$, in particular for $F = K_n$. These results extend earlier work of Erdős, Faudree, Rousseau and Schelp and of Balister, Schelp and Simonovits on so-called Ramsey size-linear graphs. Among others, we show that if $H$ is a subdivision of $K_4$ with at least $6$ vertices, then $R(H,F) = O(v(F) + e(F))$ for every graph $F$. We also conjecture that if $H$ is a connected graph with $e(H) - v(H) \leq \binom{k+1}{2} - 2$, then $R(H,K_n) = O(n^k)$. The case $k=2$ was proved by Erdős, Faudree, R…
A very nice result of Bárány and Lehel asserts that every finite subset $X$ or $\mathbb R^d$ can be covered by $f(d)$ $X$-boxes (i.e. each box has two antipodal points in $X$). As shown by Gyárfás and Pálvőlgyi this result would follow from the following conjecture : If a tournament admits a partition of its arc set into $k$ quasi orders, then its domination number is bounded in terms of $k$. This question is in turn implied by the Erdős-Sands-Sauer-Woodrow conjecture : If the arcs of a tournament $T$ are colored with $k$ colors, there is a set $X$ of at most $g(k)$ vertices such that for ever…
Is there some absolute $\varepsilon > 0$ such that for any claw-free graph $G$, the chromatic number of the square of $G$ satisfies $χ(G^2) \le (2-\varepsilon) ω(G)^2$, where $ω(G)$ is the clique number of $G$? Erdős and Nešetřil asked this question for the specific case of $G$ the line graph of a simple graph and this was answered in the affirmative by Molloy and Reed. We show that the answer to the more general question is also yes, and moreover that it essentially reduces to the original question of Erdős and Nešetřil.
We investigate the following generalisation of the 'multiplication table problem' of Erdős: given a bipartite graph with $m$ edges, how large is the set of sizes of its induced subgraphs? Erdős's problem of estimating the number of distinct products $ab$ with $a,b \le n$ is precisely the problem under consideration when the graph in question is the complete bipartite graph $K_{n,n}$. In this note, we prove that the set of sizes of the induced subgraphs of any bipartite graph with $m$ edges contains $Ω(m/(\log m)^{12})$ distinct elements.
A $k$-uniform family of subsets of $[n]$ is intersecting if it does not contain a disjoint pair of sets. The study of intersecting families is central to extremal set theory, dating back to the seminal Erdős-Ko-Rado theorem of 1961 that bounds the size of the largest such families. A recent trend has been to investigate the structure of set families with few disjoint pairs. Friedgut and Regev proved a general removal lemma, showing that when $γn \le k \le (\tfrac12 - γ)n$, a set family with few disjoint pairs can be made intersecting by removing few sets. We provide a simple proof of a removal…
