Sums, products, and exponents in two-colorings of the naturals
We prove that for any coloring of the naturals using two colors there are monochromatic sets of the form $\{x,y,xy,x+iy:i\leq k\}$ and $\{x,y,x^y,xy^i:i\leq k\}$ for any $k$.
math.CO↗
arXiv subjects
Ryan Alweiss
Publications and source records attributed to Ryan Alweiss.
$2$-large sets are sets of Bohr recurrence
Let $\alpha_1, \cdots, \alpha_d$ be real numbers, and let $S$ be the set of integers $s$ so that $||\alpha_i s||_{\mathbb{R}/\mathbb{Z}}>\delta$ for some $i$ and some fixed $\delta>0$. We prove $S$ is not \enquote{$2$-large}, i.e. there is a $2$-coloring of $\mathbb{N}$ that avoids arbitrarily long arithmetic progressions with common differences in $S$.
math.CO↗
New Obstacles to Multiple Recurrence
We show that there is a set which is not a set of multiple recurrence despite being a set of recurrence for nil-Bohr sets. This answers Huang, Shao, and Ye's \enquote{higher-order} version of Katznelson's Question on Bohr recurrence and topological recurrence in the negative. Equivalently, we construct a set $S$ so that there is a finite coloring of $\mathbb{N}$ without three-term arithmetic progressions with common differences in $S$, but so that $S$ lacks the usual polynomial obstacles to arithmetic progressions.
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Monochromatic Sums and Products over $\mathbb{Q}$
Hindman's finite sums theorem states that in any finite coloring of the naturals, there is an infinite sequence so that all of its finite subset sums are the same color. In 1979, Hindman showed that there is a finite coloring of the naturals so that no infinite sequence has all of its pairwise sums and pairwise products the same color. Hindman conjectured that for any $n$, a finite coloring of the naturals contains $n$ numbers all of whose subset sums and subset products are the same color. In this paper we prove the version of this statement where we color the rationals instead of the integers. In other words, we show that the pattern $\{ \sum_{i \in S}x_i, \prod_{i \in S}x_i \}$, where $S$ ranges over all nonempty subsets of $[n]$, is partition regular over the rationals.
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Improved Lower Bound for Frankl's Union-Closed Sets Conjecture
We verify an explicit inequality conjectured recently by Gilmer, thus proving that for any nonempty union-closed family $F \subseteq 2^{[n]}$, some $i\in [n]$ is contained in at least a $\frac{3-\sqrt{5}}{2} \approx 0.38$ fraction of the sets in $F$. One case, an explicit one-variable inequality, is checked by computer calculation.
math.CO↗
Monochromatic Sums and Products of Polynomials
We show that the pattern $\{x,x+y,xy\}$ is partition regular over the space of formal integer polynomials of degree at least one with zero constant term, with primitive recursive bounds. This provides a new proof for the partition regularity of $\{x,x+y,xy\}$ over $\mathbb{N}$, which gives the first primitive recursive bound.
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Improved bounds for the sunflower lemma
A sunflower with $r$ petals is a collection of $r$ sets so that the intersection of each pair is equal to the intersection of all of them. Erdős and Rado proved the sunflower lemma: for any fixed $r$, any family of sets of size $w$, with at least about $w^w$ sets, must contain a sunflower with $r$ petals. The famous sunflower conjecture states that the bound on the number of sets can be improved to $c^w$ for some constant $c$. In this paper, we improve the bound to about $(\log w)^w$. In fact, we prove the result for a robust notion of sunflowers, for which the bound we obtain is sharp up to lower order terms.
math.CO↗
Arithmetic Progressions in Sumsets of Sparse Sets
A set of positive integers $A \subset \mathbb{Z}_{> 0}$ is \emph{log-sparse} if there is an absolute constant $C$ so that for any positive integer $x$ the sequence contains at most $C$ elements in the interval $[x,2x)$. In this note we study arithmetic progressions in sums of log-sparse subsets of $\mathbb{Z}_{> 0}$. We prove that for any log-sparse subsets $S_1, \dots, S_n$ of $\mathbb{Z}_{> 0},$ the sumset $S = S_1 + \cdots + S_n$ cannot contain an arithmetic progression of size greater than $n^{(1+o(1))n}.$ We also show that this is nearly tight by proving that there exist log-sparse sets $S_1, \dots, S_n$ such that $S_1 + \cdots + S_n$ contains an arithmetic progression of size $n^{(1-o(1)) n}.$
math.CO↗
On a Question of Gowers on Clique Differences
We solve a question of Gowers from 2009 on clique differences in chains, thus ruling out any Sperner-type proof of the polynomial density Hales-Jewett theorem for alphabets of size 2.
math.CO↗
Discrepancy Minimization via a Self-Balancing Walk
We study discrepancy minimization for vectors in $\mathbb{R}^n$ under various settings. The main result is the analysis of a new simple random process in multiple dimensions through a comparison argument. As corollaries, we obtain bounds which are tight up to logarithmic factors for several problems in online vector balancing posed by Bansal, Jiang, Singla, and Sinha (STOC 2020), as well as linear time algorithms for logarithmic bounds for the Komlós conjecture.
cs.DS↗
On the subgraph query problem
Given a fixed graph $H$, a real number $p\in(0,1)$, and an infinite Erdős-Rényi graph $G\sim G(\infty,p)$, how many adjacency queries do we have to make to find a copy of $H$ inside $G$ with probability $1/2$? Determining this number $f(H,p)$ is a variant of the {\it subgraph query problem} introduced by Ferber, Krivelevich, Sudakov, and Vieira. For every graph $H$, we improve the trivial upper bound of $f(H,p) = O(p^{-d})$, where $d$ is the degeneracy of $H$, by exhibiting an algorithm that finds a copy of $H$ in time $o(p^{-d})$ as $p$ goes to $0$. Furthermore, we prove that there are $2$-degenerate graphs which require $p^{-2+o(1)}$ queries, showing for the first time that there exist graphs $H$ for which $f(H,p)$ does not grow like a constant power of $p^{-1}$ as $p$ goes to $0$. Finally, we answer a question of Feige, Gamarnik, Neeman, Rácz, and Tetali by showing that for any $δ< 2$, there exists $α< 2$ such that one cannot find a clique of order $α\log_2 n$ in $G(n,1/2)$ in $n^δ$ queries.
math.CO↗
Set System Blowups
We prove that given a constant $k \ge 2$ and a large set system $\mathcal{F}$ of sets of size at most $w$, a typical $k$-tuple of sets $(S_1, \cdots, S_k)$ from $\mathcal{F}$ can be ``blown up" in the following sense: for each $1 \le i \le k$, we can find a large subfamily $\mathcal{F}_i$ containing $S_i$ so that for $i \neq j$, if $T_i \in \mathcal{F}_i$ and $T_j \in \mathcal{F}_j$ , then $T_i \cap T_j=S_i \cap S_j$. We also show that the answer to the multicolor version of the sunflower conjecture is the same as the answer for the original, up to an exponential factor.
math.CO↗
On the product dimension of clique factors
The product dimension of a graph $G$ is the minimum possible number of proper vertex colorings of $G$ so that for every pair $u,v$ of non-adjacent vertices there is at least one coloring in which $u$ and $v$ have the same color. What is the product dimension $Q(s,r)$ of the vertex disjoint union of $r$ cliques, each of size $s$? Lovász, Nešetřil and Pultr proved in 1980 that for $s=2$ it is $(1+o(1)) \log_2 r$ and raised the problem of estimating this function for larger values of $s$. We show that for every fixed $s$, the answer is still $(1+o(1)) \log_2 r$ where the $o(1)$ term tends to $0$ as $r$ tends to infinity, but the problem of determining the asymptotic behavior of $Q(s,r)$ when $s$ and $r$ grow together remains open. The proof combines linear algebraic tools with the method of Gargano, Körner, and Vaccaro on Sperner capacities of directed graphs.
math.CO↗
Noisy Corruption Detection
We answer a question of Alon, Mossel, and Pemantle about the corruption detection model on graphs in the noisy setting.
math.CO↗
Ramsey Numbers of Odd Cycles Versus Larger Even Wheels
The generalized Ramsey number $R(G_1, G_2)$ is the smallest positive integer $N$ such that any red-blue coloring of the edges of the complete graph $K_N$ either contains a red copy of $G_1$ or a blue copy of $G_2$. Let $C_m$ denote a cycle of length $m$ and $W_n$ denote a wheel with $n+1$ vertices. In 2014, Zhang, Zhang and Chen determined many of the Ramsey numbers $R(C_{2k+1}, W_{n})$ of odd cycles versus larger wheels, leaving open the particular case where $n = 2j$ is even and $k<j<3k/2$. They conjectured that for these values of $j$ and $k$, $R(C_{2k+1}, W_{2j})=4j+1$. In 2015, Sanhueza-Matamala confirmed this conjecture asymptotically, showing that $R(C_{2k+1}, W_{2j}) \le 4j+334$. In this paper, we prove the conjecture of Zhang, Zhang and Chen for almost all of the remaining cases. In particular, we prove that $R(C_{2k+1},W_{2j})=4j+1$ if $j-k \ge 251$, $k<j<3k/2$, and $j \ge 212299$.
math.CO↗
Bounded gaps between primes in short intervals
Baker, Harman, and Pintz showed that a weak form of the Prime Number Theorem holds in intervals of the form $[x-x^{0.525},x]$ for large $x$. In this paper, we extend a result of Maynard and Tao concerning small gaps between primes to intervals of this length. More precisely, we prove that for any $δ\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^δ,x]$ contains $\gg_{k} \frac{x^δ}{(\log x)^k}$ pairs of consecutive primes differing by at most $d$. This confirms a speculation of Maynard that results on small gaps between primes can be refined to the setting of short intervals of this length.
math.NT↗
Asymptotic results on Klazar set partition avoidance
We establish asymptotic bounds for the number of partitions of $[n]$ avoiding a given partition in Klazar's sense, obtaining the correct answer to within an exponential for the block case. This technique also enables us to establish a general lower bound. Additionally, we consider a graph theoretic restatement of partition avoidance problems, and propose several conjectures.
math.CO↗