arXiv subjects

Jacob Fox

Publications and source records attributed to Jacob Fox.

At least 91 records · Page 5Linked to original sources

arXiv2020

A Completion of the Proof of the Edge-statistics Conjecture

For given integers $k$ and $\ell$ with $0<\ell< {k \choose 2}$, Alon, Hefetz, Krivelevich and Tyomkyn formulated the following conjecture: When sampling a $k$-vertex subset uniformly at random from a very large graph $G$, then the probability to have exactly $\ell$ edges within the sampled $k$-vertex subset is at most $e^{-1}+o_k(1)$. This conjecture was proved in the case $Ω(k)\leq \ell\leq {k \choose 2}-Ω(k)$ by Kwan, Sudakov and Tran. In this paper, we complete the proof of the conjecture by resolving the remaining cases. We furthermore give nearly tight upper bounds for the probability described above in the case $ω(1)\leq \ell\leq o(k)$. We also extend some of our results to hypergraphs with bounded edge size.

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arXiv2019

Tomescu's graph coloring conjecture for $\ell$-connected graphs

Let $P_G(k)$ be the number of proper $k$-colorings of a finite simple graph $G$. Tomescu's conjecture, which was recently solved by Fox, He, and Manners, states that $P_G(k) \le k!(k-1)^{n-k}$ for all connected graphs $G$ on $n$ vertices with chromatic number $k\geq 4$. In this paper, we study the same problem with the additional constraint that $G$ is $\ell$-connected. For $2$-connected graphs $G$, we prove a tight bound \[ P_G(k) \le (k-1)!((k-1)^{n-k+1} + (-1)^{n-k}), \] and show that equality is only achieved if $G$ is a $k$-clique with an ear attached. For $\ell \ge 3$, we prove an asymptotically tight upper bound \[ P_G(k) \le k!(k-1)^{n-\ell - k + 1} + O((k-2)^n), \] and provide a matching lower bound construction. For the ranges $k \geq \ell$ or $\ell \geq (k-2)(k-1)+1$ we further find the unique graph maximizing $P_G(k)$. We also consider generalizing $\ell$-connected graphs to connected graphs with minimum degree $δ$.

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arXiv2019

Bounded VC-dimension implies the Schur-Erdos conjecture

In 1916, Schur introduced the Ramsey number $r(3;m)$, which is the minimum integer $n$ such that for any $m$-coloring of the edges of the complete graph $K_n$, there is a monochromatic copy of $K_3$. He showed that $r(3;m) \leq O(m!)$, and a simple construction demonstrates that $r(3;m) \geq 2^{Ω(m)}$. An old conjecture of Erd\H os states that $r(3;m) = 2^{Θ(m)}$. In this note, we prove the conjecture for $m$-colorings with bounded VC-dimension, that is, for $m$-colorings with the property that the set system $\mathcal{F}$ induced by the neighborhoods of the vertices with respect to each color class has bounded VC-dimension.

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arXiv2019

Popular progression differences in vector spaces II

Green used an arithmetic analogue of Szemerédi's celebrated regularity lemma to prove the following strengthening of Roth's theorem in vector spaces. For every $α>0$, $β<α^3$, and prime number $p$, there is a least positive integer $n_p(α,β)$ such that if $n \geq n_p(α,β)$, then for every subset of $\mathbb{F}_p^n$ of density at least $α$ there is a nonzero $d$ for which the density of three-term arithmetic progressions with common difference $d$ is at least $β$. We determine for $p \geq 19$ the tower height of $n_p(α,β)$ up to an absolute constant factor and an additive term depending only on $p$. In particular, if we want half the random bound (so $β=α^3/2$), then the dimension $n$ required is a tower of twos of height $Θ\left((\log p) \log \log (1/α)\right)$. It turns out that the tower height in general takes on a different form in several different regions of $α$ and $β$, and different arguments are used both in the upper and lower bounds to handle these cases.

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arXiv2019

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.

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arXiv2019

Books versus triangles at the extremal density

A celebrated result of Mantel shows that every graph on $n$ vertices with $\lfloor n^2/4 \rfloor + 1$ edges must contain a triangle. A robust version of this result, due to Rademacher, says that there must in fact be at least $\lfloor n/2 \rfloor$ triangles in any such graph. Another strengthening, due to the combined efforts of many authors starting with Erdős, says that any such graph must have an edge which is contained in at least $n/6$ triangles. Following Mubayi, we study the interplay between these two results, that is, between the number of triangles in such graphs and their book number, the largest number of triangles sharing an edge. Among other results, Mubayi showed that for any $1/6 \leq β< 1/4$ there is $γ> 0$ such that any graph on $n$ vertices with at least $\lfloor n^2/4\rfloor + 1$ edges and book number at most $βn$ contains at least $(γ-o(1))n^3$ triangles. He also asked for a more precise estimate for $γ$ in terms of $β$. We make a conjecture about this dependency and prove this conjecture for $β= 1/6$ and for $0.2495 \leq β< 1/4$, thereby answering Mubayi's question in these ranges.

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arXiv2019

On edge-ordered Ramsey numbers

An edge-ordered graph is a graph with a linear ordering of its edges. Two edge-ordered graphs are equivalent if their is an isomorphism between them preserving the ordering of the edges. The edge-ordered Ramsey number $r_{edge}(H; q)$ of an edge-ordered graph $H$ is the smallest $N$ such that there exists an edge-ordered graph $G$ on $N$ vertices such that, for every $q$-coloring of the edges of $G$, there is a monochromatic subgraph of $G$ equivalent to $H$. Recently, Balko and Vizer announced that $r_{edge}(H;q)$ exists. However, their proof uses the Graham-Rothschild theorem and consequently gives an enormous upper bound on these numbers. We give a new proof giving a much better bound. We prove that for every edge-ordered graph $H$ on $n$ vertices, we have $r_{edge}(H;q) \leq 2^{c^qn^{2q-2}\log^q n}$, where $c$ is an absolute constant. We also explore the edge-ordered Ramsey number of sparser graphs and prove a polynomial bound for edge-ordered graphs of bounded degeneracy. We also prove a strengthening for edge-labeled graphs, graphs where every edge is given a label and the labels do not necessary have an ordering.

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arXiv2019

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.

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arXiv2019

Hypergraph cuts above the average

An r-cut of a k-uniform hypergraph H is a partition of the vertex set of H into r parts and the size of the cut is the number of edges which have a vertex in each part. A classical result of Edwards says that every m-edge graph has a 2-cut of size $m/2 + Ω(\sqrt{m})$, and this is best possible. That is, there exist cuts which exceed the expected size of a random cut by some multiple of the standard deviation. We study analogues of this and related results in hypergraphs. First, we observe that similarly to graphs, every m-edge k-uniform hypergraph has an r-cut whose size is $Ω(\sqrt m)$ larger than the expected size of a random r-cut. Moreover, in the case where k=3 and r=2 this bound is best possible and is attained by Steiner triple systems. Surprisingly, for all other cases (that is, if $k \geq 4$ or $r \geq 3$), we show that every m-edge k-uniform hypergraph has an r-cut whose size is $Ω(m^{5/9})$ larger than the expected size of a random r-cut. This is a significant difference in behaviour, since the amount by which the size of the largest cut exceeds the expected size of a random cut is now considerably larger than the standard deviation.

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arXiv2019

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.

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arXiv2019

On Ramsey numbers of hedgehogs

The hedgehog $H_t$ is a 3-uniform hypergraph on vertices $1,\dots,t+\binom{t}{2}$ such that, for any pair $(i,j)$ with $1\le i t$ such that $\{i,j,k\}$ is an edge. Conlon, Fox, and Rödl proved that the two-color Ramsey number of the hedgehog grows polynomially in the number of its vertices, while the four-color Ramsey number grows exponentially in the number of its vertices. They asked whether the two-color Ramsey number of the hedgehog $H_t$ is nearly linear in the number of its vertices. We answer this question affirmatively, proving that $r(H_t) = O(t^2\ln t)$.

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arXiv2019

Independent arithmetic progressions

We show that there is a positive constant $c$ such that any graph on vertex set $[n]$ with at most $c n^2/k^2 \log k$ edges contains an independent set of order $k$ whose vertices form an arithmetic progression. We also present applications of this result to several questions in Ramsey theory.

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arXiv2018

Semi-algebraic colorings of complete graphs

We consider $m$-colorings of the edges of a complete graph, where each color class is defined semi-algebraically with bounded complexity. The case $m = 2$ was first studied by Alon et al., who applied this framework to obtain surprisingly strong Ramsey-type results for intersection graphs of geometric objects and for other graphs arising in computational geometry. Considering larger values of $m$ is relevant, e.g., to problems concerning the number of distinct distances determined by a point set. For $p\ge 3$ and $m\ge 2$, the classical Ramsey number $R(p;m)$ is the smallest positive integer $n$ such that any $m$-coloring of the edges of $K_n$, the complete graph on $n$ vertices, contains a monochromatic $K_p$. It is a longstanding open problem that goes back to Schur (1916) to decide whether $R(p;m)=2^{O(m)}$, for a fixed $p$. We prove that this is true if each color class is defined semi-algebraically with bounded complexity. The order of magnitude of this bound is tight. Our proof is based on the Cutting Lemma of Chazelle {\em et al.}, and on a Szemerédi-type regularity lemma for multicolored semi-algebraic graphs, which is of independent interest. The same technique is used to address the semi-algebraic variant of a more general Ramsey-type problem of Erdős and Shelah.

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arXiv2018

Tower-type bounds for unavoidable patterns in words

A word $w$ is said to contain the pattern $P$ if there is a way to substitute a nonempty word for each letter in $P$ so that the resulting word is a subword of $w$. Bean, Ehrenfeucht and McNulty and, independently, Zimin characterised the patterns $P$ which are unavoidable, in the sense that any sufficiently long word over a fixed alphabet contains $P$. Zimin's characterisation says that a pattern is unavoidable if and only if it is contained in a Zimin word, where the Zimin words are defined by $Z_1 = x_1$ and $Z_n=Z_{n-1} x_n Z_{n-1}$. We study the quantitative aspects of this theorem, obtaining essentially tight tower-type bounds for the function $f(n,q)$, the least integer such that any word of length $f(n, q)$ over an alphabet of size $q$ contains $Z_n$. When $n = 3$, the first non-trivial case, we determine $f(n,q)$ up to a constant factor, showing that $f(3,q) = Θ(2^q q!)$.

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arXiv2018

Online Ramsey Numbers and the Subgraph Query Problem

The $(m,n)$-online Ramsey game is a combinatorial game between two players, Builder and Painter. Starting from an infinite set of isolated vertices, Builder draws an edge on each turn and Painter immediately paints it red or blue. Builder's goal is to force Painter to create either a red $K_m$ or a blue $K_n$ using as few turns as possible. The online Ramsey number $\tilde{r}(m,n)$ is the minimum number of edges Builder needs to guarantee a win in the $(m,n)$-online Ramsey game. By analyzing the special case where Painter plays randomly, we obtain an exponential improvement \[ \tilde{r}(n,n) \ge 2^{(2-\sqrt{2})n + O(1)} \] for the lower bound on the diagonal online Ramsey number, as well as a corresponding improvement \[ \tilde{r}(m,n) \ge n^{(2-\sqrt{2})m + O(1)} \] for the off-diagonal case, where $m\ge 3$ is fixed and $n\rightarrow\infty$. Using a different randomized Painter strategy, we prove that $\tilde{r}(3,n)=\tildeΘ(n^3)$, determining this function up to a polylogarithmic factor. We also improve the upper bound in the off-diagonal case for $m \geq 4$. In connection with the online Ramsey game with a random Painter, we study the problem of finding a copy of a target graph $H$ in a sufficiently large unknown Erdős--Rényi random graph $G(N,p)$ using as few queries as possible, where each query reveals whether or not a particular pair of vertices are adjacent. We call this problem the Subgraph Query Problem. We determine the order of the number of queries needed for complete graphs up to five vertices and prove general bounds for this problem.

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arXiv2018

A proof of Tomescu's graph coloring conjecture

In 1971, Tomescu conjectured that every connected graph $G$ on $n$ vertices with chromatic number $k\geq4$ has at most $k!(k-1)^{n-k}$ proper $k$-colorings. Recently, Knox and Mohar proved Tomescu's conjecture for $k=4$ and $k=5$. In this paper, we complete the proof of Tomescu's conjecture for all $k\ge 4$, and show that equality occurs if and only if $G$ is a $k$-clique with trees attached to each vertex.

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arXiv2018

Towards the linear arboricity conjecture

The linear arboricity of a graph $G$, denoted by $\text{la}(G)$, is the minimum number of edge-disjoint linear forests (i.e. forests in which every connected component is a path) in $G$ whose union covers all the edges of $G$. A famous conjecture due to Akiyama, Exoo, and Harary from 1980 asserts that $\text{la}(G)\leq \lceil (Δ(G)+1)/2 \rceil$, where $Δ(G)$ denotes the maximum degree of $G$. This conjectured upper bound would be best possible, as is easily seen by taking $G$ to be a regular graph. In this paper, we show that for every graph $G$, $\text{la}(G)\leq \fracΔ{2}+O(Δ^{2/3-α})$ for some $α> 0$, thereby improving the previously best known bound due to Alon and Spencer from 1992. For graphs which are sufficiently good spectral expanders, we give even better bounds. Our proofs of these results further give probabilistic polynomial time algorithms for finding such decompositions into linear forests.

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arXiv2018

A polynomial bound for the arithmetic $k$-cycle removal lemma in vector spaces

For each $k\geq 3$, Green proved an arithmetic $k$-cycle removal lemma for any abelian group $G$. The best known bounds relating the parameters in the lemma for general $G$ are of tower-type. For $k>3$, even in the case $G=\mathbb{F}_2^n$ no better bounds were known prior to this paper. This special case has received considerable attention due to its close connection to property testing of boolean functions. For every $k\geq 3$, we prove a polynomial bound relating the parameters for $G=\mathbb{F}_p^n$, where $p$ is any fixed prime. This extends the result for $k=3$ by the first two authors. Due to substantial issues with generalizing the proof of the $k=3$ case, a new strategy is developed in order to prove the result for $k>3$.

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