arXiv subjects

Sammy Luo

Publications and source records attributed to Sammy Luo.

At least 19 recordsLinked to original sources

arXiv2025

Iterated sumset expansion in $\mathbb{F}_p^n$

Given a set $A \subseteq \mathbb{F}_p^n$, what conditions does one need to guarantee that iterated sumsets of the form $A+\cdots+A$ expand quickly (say, within $O(p)$ terms) to the whole space? When only the size of $A$ is known, such expansion results are only possible when $|A|>\frac{1}{p}|\mathbb{F}_p^n|$. However, heuristic considerations suggest that expansion should begin with much smaller sets under just mild ``nondegeneracy'' conditions. In this paper, we confirm this intuition by showing a sufficient algebraic condition for the asymmetric version of this problem: We have $A_1+\dots+A_m=\mathbb{F}_p^n$ as long as each $A_i$ is not contained in the zero set of any low degree polynomial ($\text{deg} = O(n)$ when $m=O(p)$). We close with a discussion of the behavior of random sets, as well as extensions of these results and connections with the Erd\H{o}s-Ginzburg-Ziv problem. Our proofs make use of the shift operator polynomial method developed by the second author.

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arXiv2025

At most 10 cylinders mutually touch: a Ramsey-theoretic approach

Littlewood asked for the maximum number $N$ of congruent infinite cylinders that can be arranged in $\mathbb{R}^3$ so that every pair touches. We improve upon the proof of the second author that $N \leq 18$ to show that $N \leq 10$. Together with the lower bound established by Boz\'oki, Lee, and R\'onyai, this shows that $N \in \{7,8,9,10\}$. Our method is based on linear algebra and Ramsey theory, and makes partial use of computer verification. We also provide a completely computer-free proof that $N \leq 12$.

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arXiv2025

Monochromatic components with many edges in random graphs

In an $r$-coloring of edges of the complete graph on $n$ vertices, how many edges are there in the largest monochromatic connected component? A construction of Gy\'arf\'as shows that for infinitely many values of $r$, there exist colorings where all monochromatic components have at most $\left(\frac{1}{r^2-r}+o(1)\right)\binom{n}{2}$ edges. Conlon, Luo, and Tyomkyn conjectured that components with at least this many edges are attainable for all $r \ge 3$. This was proven by Luo for $r=3$, along with a lower bound of $\frac{1}{r^2-r+\frac54}{n\choose 2}$ for all $r\ge 2$, and by Conlon, Luo, and Tyomkyn for $r=4$. In this paper, we look at extensions of this problem where the graph being $r$-colored is a sparse random graph or a graph of high minimum degree. By extending several intermediate technical results from previous work in the complete graph setting, we prove analogues of the bound for general $r$ in both the sparse random setting and the high minimum degree setting, as well as the bound for $r=3$ in the latter setting.

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arXiv2024

On off-diagonal $F$-Ramsey numbers

A graph is $(t_1, t_2)$-Ramsey if any red-blue coloring of its edges contains either a red copy of $K_{t_1}$ or a blue copy of $K_{t_2}$. The size Ramsey number is the minimum number of edges contained in a $(t_1,t_2)$-Ramsey graph. Generalizing the notion of size Ramsey numbers, the $F$-Ramsey number $r_F(t_1, t_2)$ is defined to be the minimum number of copies of $F$ in a $(t_1,t_2)$-Ramsey graph. It is easy to see that $r_{K_s}(t_1,t_2)\le \binom{r(t_1,t_2)}{s}$. Recently, Fox, Tidor, and Zhang showed that equality holds in this bound when $s=3$ and $t_1=t_2$, i.e. $r_{K_3}(t,t) = \binom{r(t,t)}{3}$. They further conjectured that $r_{K_s}(t,t)=\binom{r(t,t)}{s}$ for all $s\le t$, in response to a question of Spiro. In this work, we study the off-diagonal variant of this conjecture: is it true that $r_{K_s}(t_1,t_2)=\binom{r(t_1,t_2)}{s}$ whenever $s\le \max(t_1,t_2)$? Harnessing the constructions used in the recent breakthrough work of Mattheus and Verstra\"ete on the asymptotics of $r(4,t)$, we show that when $t_1$ is $3$ or $4$, the above equality holds up to a lower order term in the exponent.

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arXiv2024

Structure of tight (k,0)-stable graphs

We say that a graph G is $(k,\ell)$-stable if removing $k$ vertices from it reduces its independence number by at most $\ell$. We say that G is tight $(k,\ell)$-stable if it is $(k,\ell)$-stable and its independence number equals $\lfloor{\frac{n-k+1}{2}\rfloor}+\ell$, the maximum possible, where $n$ is the vertex number of G. Answering a question of Dong and Wu, we show that every tight $(2,0)$-stable graph with odd vertex number must be an odd cycle. Moreover, we show that for all $k\geq 3$, every tight $(k,0)$-stable graph has at most $k+6$ vertices.

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arXiv2023

A new shift operator-based polynomial method in additive combinatorics

We introduce a new form of the polynomial method based on what we call "shift operators," which we use to give efficient and intuitive new proofs of results previously shown using a wide range of polynomial methods, including Alon's Combinatorial Nullstellensatz and the Croot-Lev-Pach method. We end by discussing some potential new directions in which the tools introduced here may be fruitfully applied.

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arXiv2023

Large monochromatic components in colorings of complete hypergraphs

Gy\'arf\'as famously showed that in every $r$-coloring of the edges of the complete graph $K_n$, there is a monochromatic connected component with at least $\frac{n}{r-1}$ vertices. A recent line of study by Conlon, Tyomkyn, and the second author addresses the analogous question about monochromatic connected components with many edges. In this paper, we study a generalization of these questions for $k$-uniform hypergraphs. Over a wide range of extensions of the definition of connectivity to higher uniformities, we provide both upper and lower bounds for the size of the largest monochromatic component that are tight up to a factor of $1+o(1)$ as the number of colors grows. We further generalize these questions to ask about counts of vertex $s$-sets contained within the edges of large monochromatic components. We conclude with more precise results in the particular case of two colors.

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arXiv2022

Small subsets with large sumset: Beyond the Cauchy--Davenport bound

For a subset $A$ of an abelian group $G$, given its size $|A|$, its doubling $\kappa=|A+A|/|A|$, and a parameter $s$ which is small compared to $|A|$, we study the size of the largest sumset $A+A'$ that can be guaranteed for a subset $A'$ of $A$ of size at most $s$. We show that a subset $A'\subseteq A$ of size at most $s$ can be found so that $|A+A'| = \Omega(\min(\kappa^{1/3},s)|A|)$. Thus a sumset significantly larger than the Cauchy--Davenport bound can be guaranteed by a bounded size subset assuming that the doubling $\kappa$ is large. Building up on the same ideas, we resolve a conjecture of Bollob\'as, Leader and Tiba that for subsets $A,B$ of $\mathbb{Z}_p$ of size at most $\alpha p$ for an appropriate constant $\alpha>0$, one only needs three elements $b_1,b_2,b_3\in B$ to guarantee $|A+\{b_1,b_2,b_3\}|\ge |A|+|B|-1$. Allowing the use of larger subsets $A'$, we show that for sets $A$ of bounded doubling, one only needs a subset $A'$ with $o(|A|)$ elements to guarantee that $A+A'=A+A$. We also address another conjecture and a question raised by Bollob\'as, Leader and Tiba on high-dimensional analogs and sets whose sumset cannot be saturated by a bounded size subset.

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arXiv2022

On random irregular subgraphs

Let $G$ be a $d$-regular graph on $n$ vertices. Frieze, Gould, Karo\'nski and Pfender began the study of the following random spanning subgraph model $H=H(G)$. Assign independently to each vertex $v$ of $G$ a uniform random number $x(v) \in [0,1]$, and an edge $(u,v)$ of $G$ is an edge of $H$ if and only if $x(u)+x(v) \geq 1$. Addressing a problem of Alon and Wei, we prove that if $d = o(n/(\log n)^{12})$, then with high probability, for each nonnegative integer $k \leq d$, there are $(1+o(1))n/(d+1)$ vertices of degree $k$ in $H$.

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arXiv2022

Monochromatic components with many edges

Given an $r$-edge-coloring of the complete graph $K_n$, what is the largest number of edges in a monochromatic connected component? This natural question has only recently received the attention it deserves, with work by two disjoint subsets of the authors resolving it for the first two special cases, when $r = 2$ or $3$. Here we introduce a general framework for studying this problem and apply it to fully resolve the $r = 4$ case, showing that any $4$-edge-coloring of $K_n$ contains a monochromatic component with at least $\frac{1}{12}\binom{n}{2}$ edges, where the constant $\frac{1}{12}$ is optimal only when the coloring matches a certain construction of Gy\'arf\'as.

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arXiv2021

On connected components with many edges

We prove that if $H$ is a subgraph of a complete multipartite graph $G$, then $H$ contains a connected component $H'$ satisfying $|E(H')||E(G)|\geq |E(H)|^2$. We use this to prove that every three-coloring of the edges of a complete graph contains a monochromatic connected subgraph with at least $1/6$ of the edges. We further show that such a coloring has a monochromatic circuit with a fraction $1/6-o(1)$ of the edges. This verifies a conjecture of Conlon and Tyomkyn. Moreover, for general $k$, we show that every $k$-coloring of the edges of $K_n$ contains a monochromatic connected subgraph with at least $\frac{1}{k^2-k+\frac{5}{4}}\binom{n}{2}$ edges.

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arXiv2021

Multicolor list Ramsey numbers grow exponentially

The list Ramsey number $R_{\ell}(H,k)$, recently introduced by Alon, Buci\'c, Kalvari, Kuperwasser, and Szab\'o, is a list-coloring variant of the classical Ramsey number. They showed that if $H$ is a fixed $r$-uniform hypergraph that is not $r$-partite and the number of colors $k$ goes to infinity, $e^{\Omega(\sqrt{k})} \le R_{\ell} (H,k) \le e^{O(k)}$. We prove that $R_{\ell}(H,k) = e^{\Theta(k)}$ if and only if $H$ is not $r$-partite.

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arXiv2019

Extremal and Ramsey results on graph blowups

Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite Ramsey numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring of $G$ contains a monochromatic copy of $H$. Let $H[t]$ denote the $t$-blowup of $H$. Then the blowup Ramsey number of $G,H,r,$ and $t$ is defined as the minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. Souza proved upper and lower bounds on $n$ that are exponential in $t$, and conjectured that the exponential constant does not depend on $G$. We prove that the dependence on $G$ in the exponential constant is indeed unnecessary, but conjecture that some dependence on $G$ is unavoidable. An important step in both Souza's proof and ours is a theorem of Nikiforov, which says that if a graph contains a constant fraction of the possible copies of $H$, then it contains a blowup of $H$ of logarithmic size. We also provide a new proof of this theorem with a better quantitative dependence.

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arXiv2019

Generalized Minors and Tensor Invariants

Berenstein, Fomin and Zelevinsky defined functions on double Bruhat cells which they called generalized minors. By relating certain double Bruhat cells to configuration spaces of flags, we give formulas for these generalized minors as tensor invariants. This allows us to verify certain weight identities. The weights of the tensor invariants can then be used to construct the quiver for the cluster structure on the configuration space of three flags. We also show a converse statement--that the weights of tensor invariants can by computed from the structure of the quiver. The weight identities are important because they are necessary for the existence of cluster structures on the moduli space of framed local systems.

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arXiv2017

Stability and exact Turan numbers for matroids

We consider the Tur\'an-type problem of bounding the size of a set $M \subseteq \mathbb{F}_2^n$ that does not contain a linear copy of a given fixed set $N \subseteq \mathbb{F}_2^k$, where $n$ is large compared to $k$. An Erd\H{o}s-Stone type theorem [5] in this setting gives a bound that is tight up to a $o(2^n)$ error term; our first main result gives a stability version of this theorem, showing that such an $M$ that is close in size to the upper bound in [5] is close in edit distance to the obvious extremal example. Our second result shows that the error term in [5] is exactly controlled by the solution to one of a class of `sparse' extremal problems, and in many cases eliminates the error term completely to give a sharp upper bound on $|M|$.

math.CO↗

arXiv2017

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 $\delta\in [0.525,1]$ there exist positive integers $k,d$ such that for sufficiently large $x$, the interval $[x-x^\delta,x]$ contains $\gg_{k} \frac{x^\delta}{(\log x)^k}$ 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.

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arXiv2016

A Counting Lemma for Binary Matroids and Applications to Extremal Problems

In graph theory, the Szemer\'edi regularity lemma gives a decomposition of the indicator function for any graph $G$ into a structured component, a uniform part, and a small error. This result, in conjunction with a counting lemma that guarantees many copies of a subgraph $H$ provided a copy of $H$ appears in the structured component, is used in many applications to extremal problems. An analogous decomposition theorem exists for functions over $\mathbb{F}_p^n$. Specializing to $p=2$, we obtain a statement about the indicator functions of simple binary matroids. In this paper we extend previous results to prove a corresponding counting lemma for binary matroids. We then apply this counting lemma to give simple proofs of some known extremal results, analogous to the proofs of their graph-theoretic counterparts, and discuss how to use similar methods to attack a problem concerning the critical numbers of dense binary matroids avoiding a fixed submatroid.

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arXiv2016

Short Zero-Sum Sequences Over Abelian $p$-Groups of Large Exponent

Let $G$ be a finite abelian group with exponent $n$. Let $\eta(G)$ denote the smallest integer $\ell$ such that every sequence over $G$ of length at least $\ell$ has a zero-sum subsequence of length at most $n$. We determine the precise value of $\eta(G)$ when $G$ is a $p$-group whose Davenport constant is at most $2n-1$. This confirms one of the equalities in a conjecture by Schmid and Zhuang from 2010.

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