Impartial digraphs
We prove a conjecture of Fox, Huang, and Lee that characterizes directed graphs that have constant density in all tournaments: they are disjoint unions of trees that are each constructed in a certain recursive way.
math.CO↗
arXiv subjects
Yufei Zhao
Publications and source records attributed to Yufei Zhao.
A reverse Sidorenko inequality
Let $H$ be a graph allowing loops as well as vertex and edge weights. We prove that, for every triangle-free graph $G$ without isolated vertices, the weighted number of graph homomorphisms $\hom(G, H)$ satisfies the inequality \[ \hom(G, H ) \le \prod_{uv \in E(G)} \hom(K_{d_u,d_v}, H )^{1/(d_ud_v)}, \] where $d_u$ denotes the degree of vertex $u$ in $G$. In particular, one has \[ \hom(G, H )^{1/|E(G)|} \le \hom(K_{d,d}, H )^{1/d^2} \] for every $d$-regular triangle-free $G$. The triangle-free hypothesis on $G$ is best possible. More generally, we prove a graphical Brascamp-Lieb type inequality, where every edge of $G$ is assigned some two-variable function. These inequalities imply tight upper bounds on the partition function of various statistical models such as the Ising and Potts models, which includes independent sets and graph colorings. For graph colorings, corresponding to $H = K_q$, we show that the triangle-free hypothesis on $G$ may be dropped; this is also valid if some of the vertices of $K_q$ are looped. A corollary is that among $d$-regular graphs, $G = K_{d,d}$ maximizes the quantity $c_q(G)^{1/|V(G)|}$ for every $q$ and $d$, where $c_q(G)$ counts proper $q$-colorings of $G$. Finally, we show that if the edge-weight matrix of $H$ is positive semidefinite, then \[ \hom(G, H) \le \prod_{v \in V(G)} \hom(K_{d_v+1}, H )^{1/(d_v+1)}. \] This implies that among $d$-regular graphs, $G = K_{d+1}$ maximizes $\hom(G, H)^{1/|V(G)|}$. For 2-spin Ising models, our results give a complete characterization of extremal graphs: complete bipartite graphs maximize the partition function of 2-spin antiferromagnetic models and cliques maximize the partition function of ferromagnetic models. These results settle a number of conjectures by Galvin-Tetali, Galvin, and Cohen-Csikvári-Perkins-Tetali, and provide an alternate proof to a conjecture by Kahn.
math.CO↗
A short proof of the canonical polynomial van der Waerden theorem
We present a short new proof of the canonical polynomial van der Waerden theorem, recently established by Girao [arXiv:2004.07766].
math.CO↗
Tower-type bounds for Roth's theorem with popular differences
Green developed an arithmetic regularity lemma to prove a strengthening of Roth's theorem on arithmetic progressions in dense sets. It states that for every $ε> 0$ there is some $N_0(ε)$ such that for every $N \ge N_0(ε)$ and $A \subset [N]$ with $|A| = αN$, there is some nonzero $d$ such that $A$ contains at least $(α^3 - ε) N$ three-term arithmetic progressions with common difference $d$. We prove that the minimum $N_0(ε)$ in Green's theorem is an exponential tower of 2s of height on the order of $\log(1/ε)$. Both the lower and upper bounds are new. It shows that the tower-type bounds that arise from the use of a regularity lemma in this application are quantitatively necessary.
math.CO↗
Spectroscopic realization of large surface gap in a doped magnetic topological insulator
Realization of the quantum anomalous Hall effect and axion electrodynamics in topological materials are among the paradigmatic phenomena in condensed matter physics. Recently, signatures of both phases are observed to exist in thin films of MnBi$_2$Te$_4$, a stoichiometric antiferromagnetic topological insulator. Direct evidence of the bulk topological magnetoelectric response in an axion insulator requires an energy gap at its topological surface state (TSS). However, independent spectroscopic experiments revealed that such a surface gap is absent, or much smaller than previously thought, in MnBi$_2$Te$_4$. Here, we utilize angle resolved photoemission spectroscopy and density functional theory calculations to demonstrate that a sizable TSS gap unexpectedly exists in Sb-doped MnBi$_2$Te$_4$. This gap is found to be topologically nontrivial, insensitive to the bulk antiferromagnetic-paramagnetic transition, while enlarges along with increasing Sb concentration. Our work shows that Mn(Bi$_{1-x}$Sb$_x$)$_2$Te$_4$ is a potential platform to observe the key features of the high-temperature axion insulator state, such as the topological magnetoelectric responses and half-integer quantum Hall effects.
cond-mat.mtrl-sci↗
Exponential improvements for superball packing upper bounds
We prove that for all fixed $p > 2$, the translative packing density of unit $\ell_p$-balls in $\mathbb{R}^n$ is at most $2^{(γ_p + o(1))n}$ with $γ_p < - 1/p$. This is the first exponential improvement in high dimensions since van der Corput and Schaake (1936).
math.MG↗
Induced arithmetic removal: complexity 1 patterns over finite fields
We prove an arithmetic analog of the induced graph removal lemma for complexity 1 patterns over finite fields. Informally speaking, we show that given a fixed collection of $r$-colored complexity 1 arithmetic patterns over $\mathbb F_q$, every coloring $ϕ\colon \mathbb F_q^n \setminus\{0\} \to [r]$ with $o(1)$ density of every such pattern can be recolored on an $o(1)$-fraction of the space so that no such pattern remains.
math.CO↗
Triforce and Corners
May the $\mathit{triforce}$ be the 3-uniform hypergraph on six vertices with edges $\{123',12'3,1'23\}$. We show that the minimum triforce density in a 3-uniform hypergraph of edge density $δ$ is $δ^{4-o(1)}$ but not $O(δ^4)$. Let $M(δ)$ be the maximum number such that the following holds: for every $ε> 0$ and $G = \mathbb{F}_2^n$ with $n$ sufficiently large, if $A \subseteq G \times G$ with $A \ge δ|G|^2$, then there exists a nonzero "popular difference" $d \in G$ such that the number of "corners" $(x,y), (x+d,y), (x,y+d) \in A$ is at least $(M(δ) - ε)|G|^2$. As a corollary via a recent result of Mandache, we conclude that $M(δ) = δ^{4-o(1)}$ and $M(δ) = ω(δ^4)$. On the other hand, for $0 < δ< 1/2$ and sufficiently large $N$, there exists $A \subseteq [N]^3$ with $|A|\geδN^3$ such that for every $d \ne 0$, the number of corners $(x,y,z), (x+d,y,z),(x,y+d,z),(x,y,z+d) \in A$ is at most $δ^{c \log (1/δ)} N^3$. A similar bound holds in higher dimensions, or for any configuration with at least 5 points or affine dimension at least 3.
math.CO↗
Efficient arithmetic regularity and removal lemmas for induced bipartite patterns
Let $G$ be an abelian group of bounded exponent and $A \subseteq G$. We show that if the collection of translates of $A$ has VC dimension at most $d$, then for every $ε>0$ there is a subgroup $H$ of $G$ of index at most $ε^{-d-o(1)}$ such that one can add or delete at most $ε|G|$ elements to/from $A$ to make it a union of $H$-cosets. We also establish a removal lemma with polynomial bounds, with applications to property testing, for induced bipartite patterns in a finite abelian group with bounded exponent.
math.CO↗
The number of independent sets in an irregular graph
Settling Kahn's conjecture (2001), we prove the following upper bound on the number $i(G)$ of independent sets in a graph $G$ without isolated vertices: \[ i(G) \le \prod_{uv \in E(G)} i(K_{d_u,d_v})^{1/(d_u d_v)}, \] where $d_u$ is the degree of vertex $u$ in $G$. Equality occurs when $G$ is a disjoint union of complete bipartite graphs. The inequality was previously proved for regular graphs by Kahn and Zhao. We also prove an analogous tight lower bound: \[ i(G) \ge \prod_{v \in V(G)} i(K_{d_v+1})^{1/(d_v + 1)}, \] where equality occurs for $G$ a disjoint union of cliques. More generally, we prove bounds on the weighted versions of these quantities, i.e., the independent set polynomial, or equivalently the partition function of the hard-core model with a given fugacity on a graph.
math.CO↗
On the number of Hadamard matrices via anti-concentration
Many problems in combinatorial linear algebra require upper bounds on the number of solutions to an underdetermined system of linear equations $Ax = b$, where the coordinates of the vector $x$ are restricted to take values in some small subset (e.g. $\{\pm 1\}$) of the underlying field. The classical ways of bounding this quantity are to use either a rank bound observation due to Odlyzko or a vector anti-concentration inequality due to Halász. The former gives a stronger conclusion except when the number of equations is significantly smaller than the number of variables; even in such situations, the hypotheses of Halász's inequality are quite hard to verify in practice. In this paper, using a novel approach to the anti-concentration problem for vector sums, we obtain new Halász-type inequalities which beat the Odlyzko bound even in settings where the number of equations is comparable to the number of variables. In addition to being stronger, our inequalities have hypotheses which are considerably easier to verify. We present two applications of our inequalities to combinatorial (random) matrix theory: (i) we obtain the first non-trivial upper bound on the number of $n\times n$ Hadamard matrices, and (ii) we improve a recent bound of Deneanu and Vu on the probability of normality of a random $\{\pm 1\}$ matrix.
math.PR↗
The Green-Tao theorem: an exposition
The celebrated Green-Tao theorem states that the prime numbers contain arbitrarily long arithmetic progressions. We give an exposition of the proof, incorporating several simplifications that have been discovered since the original paper.
math.NT↗
Upper tails for arithmetic progressions in a random set
Let $X_k$ denote the number of $k$-term arithmetic progressions in a random subset of $\mathbb{Z}/N\mathbb{Z}$ or $\{1, \dots, N\}$ where every element is included independently with probability $p$. We determine the asymptotics of $\log \mathbb{P}(X_k \ge (1+δ) \mathbb{E} X_k)$ (also known as the large deviation rate) where $p \to 0$ with $p \ge N^{-c_k}$ for some constant $c_k > 0$, which answers a question of Chatterjee and Dembo. The proofs rely on the recent nonlinear large deviation principle of Eldan, which improved on earlier results of Chatterjee and Dembo. Our results complement those of Warnke, who used completely different methods to estimate, for the full range of $p$, the large deviation rate up to a constant factor.
math.PR↗
On regularity lemmas and their algorithmic applications
Szemerédi's regularity lemma and its variants are some of the most powerful tools in combinatorics. In this paper, we establish several results around the regularity lemma. First, we prove that whether or not we include the condition that the desired vertex partition in the regularity lemma is equitable has a minimal effect on the number of parts of the partition. Second, we use an algorithmic version of the (weak) Frieze--Kannan regularity lemma to give a substantially faster deterministic approximation algorithm for counting subgraphs in a graph. Previously, only an exponential dependence for the running time on the error parameter was known, and we improve it to a polynomial dependence. Third, we revisit the problem of finding an algorithmic regularity lemma, giving approximation algorithms for several co-NP-complete problems. We show how to use the weak Frieze--Kannan regularity lemma to approximate the regularity of a pair of vertex subsets. We also show how to quickly find, for each $ε'>ε$, an $ε'$-regular partition with $k$ parts if there exists an $ε$-regular partition with $k$ parts. Finally, we give a simple proof of the permutation regularity lemma which improves the tower-type bound on the number of parts in the previous proofs to a single exponential bound.
math.CO↗
A fast new algorithm for weak graph regularity
We provide a deterministic algorithm that finds, in $ε^{-O(1)} n^2$ time, an $ε$-regular Frieze-Kannan partition of a graph on $n$ vertices. The algorithm outputs an approximation of a given graph as a weighted sum of $ε^{-O(1)}$ many complete bipartite graphs. As a corollary, we give a deterministic algorithm for estimating the number of copies of $H$ in an $n$-vertex graph $G$ up to an additive error of at most $εn^{v(H)}$, in time $ε^{-O_H(1)}n^2$.
math.CO↗
Upper tails and independence polynomials in random graphs
The upper tail problem in the Erdős--Rényi random graph $G\sim\mathcal{G}_{n,p}$ asks to estimate the probability that the number of copies of a graph $H$ in $G$ exceeds its expectation by a factor $1+δ$. Chatterjee and Dembo showed that in the sparse regime of $p\to 0$ as $n\to\infty$ with $p \geq n^{-α}$ for an explicit $α=α_H>0$, this problem reduces to a natural variational problem on weighted graphs, which was thereafter asymptotically solved by two of the authors in the case where $H$ is a clique. Here we extend the latter work to any fixed graph $H$ and determine a function $c_H(δ)$ such that, for $p$ as above and any fixed $δ>0$, the upper tail probability is $\exp[-(c_H(δ)+o(1))n^2 p^Δ\log(1/p)]$, where $Δ$ is the maximum degree of $H$. As it turns out, the leading order constant in the large deviation rate function, $c_H(δ)$, is governed by the independence polynomial of $H$, defined as $P_H(x)=\sum i_H(k) x^k $ where $i_H(k)$ is the number of independent sets of size $k$ in $H$. For instance, if $H$ is a regular graph on $m$ vertices, then $c_H(δ)$ is the minimum between $\frac12 δ^{2/m}$ and the unique positive solution of $P_H(x) = 1+δ$.
math.CO↗
Group representations that resist worst-case sampling
Motivated by expansion in Cayley graphs, we show that there exist infinitely many groups $G$ with a nontrivial irreducible unitary representation whose average over every set of $o(\log\log|G|)$ elements of $G$ has operator norm $1 - o(1)$. This answers a question of Lovett, Moore, and Russell, and strengthens their negative answer to a question of Wigderson. The construction is the affine group of $\mathbb{F}_p$ and uses the fact that for every $A \subset \mathbb{F}_p\setminus\{0\}$, there is a set of size $\exp(\exp(O(|A|)))$ that is almost invariant under both additive and multiplicatpive translations by elements of $A$.
math.CO↗
Extremal regular graphs: independent sets and graph homomorphisms
This survey concerns regular graphs that are extremal with respect to the number of independent sets, and more generally, graph homomorphisms. More precisely, in the family of of $d$-regular graphs, which graph $G$ maximizes/minimizes the quantity $i(G)^{1/v(G)}$, the number of independent sets in $G$ normalized exponentially by the size of $G$? What if $i(G)$ is replaced by some other graph parameter? We review existing techniques, highlight some exciting recent developments, and discuss open problems and conjectures for future research.
math.CO↗