Search

A curve in the plane is $x$-monotone if every vertical line intersects it at most once. A family of curves are called pseudo-segments if every pair of them have at most one point in common. We construct $2^{\Omega(n^{4/3})}$ families, each consisting of $n$ labelled $x$-monotone pseudo-segments such that their intersection graphs are different. On the other hand, we show that the number of such intersection graphs is at most $2^{O(n^{3/2-\varepsilon})}$, where $\varepsilon>0$ is a suitable constant. Our proof uses a new upper bound on the number of set systems of size $m$ on a ground set of size $n$, with VC-dimension at most $d$. Much better upper bounds are obtained if we only count bipartite intersection graphs, or, in general, intersection graphs with bounded chromatic number.

We initiate the systematic study of the following Tur\'an-type question. Suppose $\Gamma$ is a graph with $n$ vertices such that the edge density between any pair of subsets of vertices of size at least $t$ is at most $1 - c$, for some $t$ and $c > 0$. What is the largest number of edges in a subgraph $G \subseteq \Gamma$ which does not contain a fixed graph $H$ as an induced subgraph or, more generally, which belongs to a hereditary property $\mathcal{P}$? This provides a common generalization of two recently studied cases, namely $\Gamma$ being a (pseudo-)random graph and a graph without a large complete bipartite subgraph. We focus on the interesting case where $H$ is a bipartite graph. We determine the answer up to a constant factor with respect to $n$ and $t$, for certain bipartite $H$ and for $\Gamma$ either a dense random graph or a Paley graph with a square number of vertices. In particular, our bounds match if $H$ is a tree, or if one part of $H$ has $d$ vertices complete to the other part, all other vertices in that part have degree at most $d$, and the other part has sufficiently many vertices. As applications of the latter result, we answer a question of Alon, Krivelevich, and Samotij on the largest subgraph with a hereditary property which misses a bipartite graph, and determine up to a constant factor the largest number of edges in a string subgraph of $\Gamma$. The proofs are based on a variant of the dependent random choice and a novel approach for finding induced copies by inductively defining probability distributions supported on induced copies of smaller subgraphs., Comment: 20 pages

Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$, $$ (3\ell - 1) \cdot 2^{n-5} < ES_{\ell}(n) < \ell^2 \cdot 2^{n+ C\sqrt{n\log n}}.$$ A similar extension of the well-known Erd\H os--Szekeres cups-caps theorem is also proved.

Answering a question of Erd\H{o}s and Graham, we show that for each fixed positive rational number $x$ the number of ways to write $x$ as a sum of reciprocals of distinct positive integers each at most $n$ is $2^{(c_x + o(1))n}$ for an explicit constant $c_x$ increasing with $x$., Comment: 8 pages

A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{\Omega_H(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear., Comment: 22 pages

It is well-known that polynomial versions of theorems of R\"odl and Nikiforov, as conjectured by Fox and Sudakov and Nguyen, Scott and Seymour imply the classical Erd\H{o}s-Hajnal conjecture. In this note, we prove that these three conjectures are in fact equivalent, extending several previous particular results in this direction by Fox, Nguyen, Scott and Seymour; Nguyen, Scott and Seymour and Gishboliner and Shapira. We deduce that the family of string graphs satisfies the polynomial R\"odl conjecture. We also derive analogous results for hypergraphs, tournaments, ordered graphs, and colored graphs., Comment: Added a section showing analogs of our results in the setting of tournaments, ordered graphs, hypergraphs and colored graphs

We study variants of Sidorenko's conjecture in tournaments, where new phenomena arise that do not have clear analogues in the setting of undirected graphs. We first consider oriented graphs that are systematically under-represented in tournaments (called tournament anti-Sidorenko). We prove that such oriented graphs must be quite sparse; specifically, the maximum number of edges of a $k$-vertex oriented graph which is tournament anti-Sidorenko is $(1+o(1))k\log_2 k$. We also give several novel constructions of oriented graphs that are systematically over-represented in tournaments (tournament Sidorenko); as a representative example, we show that most ways to delete an edge from a transitive tournament yield a tournament Sidorenko oriented graph. As an illustration of our methods, we characterize which orientations of stars are tournament Sidorenko and which are tournament anti-Sidorenko., Comment: 23 pages, 2 figures

We prove that every partially ordered set on $n$ elements contains $k$ subsets $A_{1},A_{2},\dots,A_{k}$ such that either each of these subsets has size $\Omega(n/k^{5})$ and, for every $i_{\ell}a_{2}>_{\ell}\dots>_{\ell}a_{k}$ for any $(a_1,a_2,\dots,a_k) \in A_1\times A_2\times \dots \times A_k$, or $a_i$ is incomparable with $a_j$ for any $i\ne j$, $a_i\in A_i$ and $a_j\in A_j$. This improves on a 2009 result of Pach and the first author motivated by problems in discrete geometry.

Degeneracy plays an important role in understanding Tur\'an- and Ramsey-type properties of graphs. Unfortunately, the usual hypergraphical generalization of degeneracy fails to capture these properties. We define the skeletal degeneracy of a $k$-uniform hypergraph as the degeneracy of its $1$-skeleton (i.e., the graph formed by replacing every $k$-edge by a $k$-clique). We prove that skeletal degeneracy controls hypergraph Tur\'an and Ramsey numbers in a similar manner to (graphical) degeneracy. Specifically, we show that $k$-uniform hypergraphs with bounded skeletal degeneracy have linear Ramsey number. This is the hypergraph analogue of the Burr-Erd\H{o}s conjecture (proved by Lee). In addition, we give upper and lower bounds of the same shape for the Tur\'an number of a $k$-uniform $k$-partite hypergraph in terms of its skeletal degeneracy. The proofs of both results use the technique of dependent random choice. In addition, the proof of our Ramsey result uses the `random greedy process' introduced by Lee in his resolution of the Burr-Erd\H{o}s conjecture., Comment: 33 pages

The $q$-color Ramsey number of a $k$-uniform hypergraph $G,$ denoted $r(G;q)$, is the minimum integer $N$ such that any coloring of the edges of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $G$. The study of these numbers is one of the most central topics in combinatorics. One natural question, which for triangles goes back to the work of Schur in 1916, is to determine the behaviour of $r(G;q)$ for fixed $G$ and $q$ tending to infinity. In this paper we study this problem for $3$-uniform hypergraphs and determine the tower height of $r(G;q)$ as a function of $q$. More precisely, given a hypergraph $G$, we determine when $r(G; q)$ behaves polynomially, exponentially or double-exponentially in $q$. This answers a question of Axenovich, Gy\'{a}rf\'{a}s, Liu and Mubayi.

A graph is $H$-Ramsey if every two-coloring of its edges contains a monochromatic copy of $H$. Define the $F$-Ramsey number of $H$, denoted by $r_F(H)$, to be the minimum number of copies of $F$ in a graph which is $H$-Ramsey. This generalizes the Ramsey number and size Ramsey number of a graph. Addressing a question of Spiro, we prove that \[r_{K_3}(K_t)=\binom{r(K_t)}{3}\] for all sufficiently large $t$. We do so through a result on graph coloring: there exists an absolute constant $K$ such that every $r$-chromatic graph where every edge is contained in at least $K$ triangles must contain at least $\binom{r}{3}$ triangles in total., Comment: 14 pages

We prove a far-reaching strengthening of Szemer\'edi's regularity lemma for intersection graphs of pseudo-segments. It shows that the vertex set of such a graph can be partitioned into a bounded number of parts of roughly the same size such that almost all bipartite graphs between different pairs of parts are complete or empty. We use this to get an improved bound on disjoint edges in simple topological graphs, showing that every $n$-vertex simple topological graph with no $k$ pairwise disjoint edges has at most $n(\log n)^{O(\log k)}$ edges.

We show that for every positive integer $k$ there are positive constants $C$ and $c$ such that if $A$ is a subset of $\{1, 2, \dots, n\}$ of size at least $C n^{1/k}$, then, for some $d \leq k-1$, the set of subset sums of $A$ contains a homogeneous $d$-dimensional generalized arithmetic progression of size at least $c|A|^{d+1}$. This strengthens a result of Szemer\'edi and Vu, who proved a similar statement without the homogeneity condition. As an application, we make progress on the Erd\H{o}s--Straus non-averaging sets problem, showing that every subset $A$ of $\{1, 2, \dots, n\}$ of size at least $n^{\sqrt{2} - 1 + o(1)}$ contains an element which is the average of two or more other elements of $A$. This gives the first polynomial improvement on a result of Erd\H{o}s and S\'ark\"ozy from 1990., Comment: 34 pages

The $q$-color Ramsey number of a $k$-uniform hypergraph $H$ is the minimum integer $N$ such that any $q$-coloring of the complete $k$-uniform hypergraph on $N$ vertices contains a monochromatic copy of $H$. The study of these numbers is one of the central topics in Combinatorics. In 1973, Erd\H{o}s and Graham asked to maximize the Ramsey number of a graph as a function of the number of its edges. Motivated by this problem, we study the analogous question for hypergaphs. For fixed $k \ge 3$ and $q \ge 2$ we prove that the largest possible $q$-color Ramsey number of a $k$-uniform hypergraph with $m$ edges is at most $\mathrm{tw}_k(O(\sqrt{m})),$ where $\mathrm{tw}$ denotes the tower function. We also present a construction showing that this bound is tight for $q \ge 4$. This resolves a problem by Conlon, Fox and Sudakov. They previously proved the upper bound for $k \geq 4$ and the lower bound for $k=3$. Although in the graph case the tightness follows simply by considering a clique of appropriate size, for higher uniformities the construction is rather involved and is obtained by using paths in expander graphs.

A well-known theorem of R\"odl says that for every graph $H$, and every $\epsilon>0$, there exists $\delta>0$ such that if $G$ does not contain an induced copy of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge\delta|G|$ such that one of $G[X],\overline{G}[X]$ has edge-density at most $\epsilon$. But how does $\delta$ depend on $\epsilon$? Fox and Sudakov conjectured that the dependence can be taken to be polynomial: for all $H$ there exists $c>0$ such that for all $\epsilon\in(0,1/2)$, R\"odl's theorem holds with $\delta=\epsilon^c$. This conjecture implies the Erd\H{o}s-Hajnal conjecture, and until now it had not been verified for any non-trivial graphs $H$. Our first result shows that it is true when $H=P_4$ (indeed, we can take $\delta=\epsilon$, and insist that one of $G[X],\overline{G}[X]$ has maximum degree at most $\epsilon^2|G|$).) We will generalize this, and to do so, we need to work with an even stronger property. Let us say $H$ is {\em viral} if there exists $c>0$ such that for all $\epsilon\in(0,1/2)$, if $G$ contains at most $\epsilon^c|G|^{|H|}$ induced copies of $H$, then there exists $X\subseteq V(G)$ with $|X|\ge \epsilon^c|G|$ such that one of $G[X],\overline{G}[X]$ has density at most $\epsilon$. We will show that $P_4$ is viral, using a ``polynomial $P_4$-removal lemma'' of Alon and Fox. We will also show that viral graphs are closed under vertex-substitution, and so all graphs that can be obtained by substitution from copies of $P_4$ are viral. (In a subsequent paper, it will be shown that all graphs currently known to satisfy the Erd\H{o}s-Hajnal conjecture are in fact viral.) Finally, we give a different strengthening of R\"odl's theorem: we show that if $G$ does not contain an induced copy of $P_4$, then its vertices can be partitioned into at most $480\epsilon^{-4}$ subsets $X$ such that one of $G[X],\overline{G}[X]$ has density at most $\epsilon$.

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is a subset of $n$ vertices where all of the edges between them receive a common color. If $n$ is fixed and $\frac{s}{r}$ is less than and bounded away from $1-\frac{1}{n-1}$, then $R(n;r,s)$ is known to grow exponentially in $r$, while if $\frac{s}{r}$ is greater than and bounded away from $1-\frac{1}{n-1}$, then $R(n;r,s)$ is bounded. Here we prove bounds for $R(n;r,s)$ in the intermediate range where $\frac{s}{r}$ is close to $1 - \frac{1}{n-1}$ by establishing a connection to the maximum size of error-correcting codes near the zero-rate threshold.

We study analogues of Sidorenko's conjecture and the forcing conjecture in oriented graphs, showing that natural variants of these conjectures in directed graphs are equivalent to the asymmetric, undirected analogues of the conjectures., Comment: 9 pages, 1 figure

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.

Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number $r(K_{4}^{(3)},S_n^{(3)})$ exhibits an unusual intermediate growth rate, namely, \[ 2^{c \log^2 n} \le r(K_{4}^{(3)},S_n^{(3)}) \le 2^{c' n^{2/3}\log n} \] for some positive constants $c$ and $c'$. The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum $N$ such that any $2$-edge-coloring of the Cartesian product $K_N \square K_N$ contains either a red rectangle or a blue $K_n$?, Comment: 13 pages

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 $H$., Comment: 18 pages

Extending an earlier conjecture of Erd\H{o}s, Burr and Rosta conjectured that among all two-colorings of the edges of a complete graph, the uniformly random coloring asymptotically minimizes the number of monochromatic copies of any fixed graph $H$. This conjecture was disproved independently by Sidorenko and Thomason. The first author later found quantitatively stronger counterexamples, using the Tur\'an coloring, in which one of the two colors spans a balanced complete multipartite graph. We prove that the Tur\'an coloring is extremal for an infinite family of graphs, and that it is the unique extremal coloring. This yields the first determination of the Ramsey multiplicity constant of a graph for which the Burr--Rosta conjecture fails. We also prove an analogous three-color result. In this case, our result is conditional on a certain natural conjecture on the behavior of two-color Ramsey numbers., Comment: 39 pages, final version to appear in Advances in Combinatorics

For positive integers $n,r,s$ with $r > s$, the set-coloring Ramsey number $R(n;r,s)$ is the minimum $N$ such that if every edge of the complete graph $K_N$ receives a set of $s$ colors from a palette of $r$ colors, then there is guaranteed to be a monochromatic clique on $n$ vertices, that is, a subset of $n$ vertices where all of the edges between them receive a common color. In particular, the case $s=1$ corresponds to the classical multicolor Ramsey number. We prove general upper and lower bounds on $R(n;r,s)$ which imply that $R(n;r,s) = 2^{\Theta(nr)}$ if $s/r$ is bounded away from $0$ and $1$. The upper bound extends an old result of Erd\H{o}s and Szemer\'edi, who treated the case $s = r-1$, while the lower bound exploits a connection to error-correcting codes. We also study the analogous problem for hypergraphs., Comment: 11 pages

A directed graph is oriented if it can be obtained by orienting the edges of a simple, undirected graph. For an oriented graph $G$, let $\beta(G)$ denote the size of a minimum feedback arc set, a smallest subset of edges whose deletion leaves an acyclic subgraph. A simple consequence of a result of Berger and Shor is that any oriented graph $G$ with $m$ edges satisfies $\beta(G) = m/2 - \Omega(m^{3/4})$. We observe that if an oriented graph $G$ has a fixed forbidden subgraph $B$, the upper bound of $\beta(G) = m/2 - \Omega(m^{3/4})$ is best possible as a function of the number of edges if $B$ is not bipartite, but the exponent $3/4$ in the lower order term can be improved if $B$ is bipartite. We also show that for every rational number $r$ between $3/4$ and $1$, there is a finite collection of digraphs $\mathcal{B}$ such that every $\mathcal{B}$-free digraph $G$ with $m$ edges satisfies $\beta(G) = m/2 - \Omega(m^r)$, and this bound is best possible up to the implied constant factor. The proof uses a connection to Tur\'an numbers and a result of Bukh and Conlon. Both of our upper bounds come equipped with randomized linear-time algorithms that construct feedback arc sets achieving those bounds. Finally, we give a characterization of quasirandom directed graphs via minimum feedback arc sets., Comment: 23 pages

An $r$-quasiplanar graph is a graph drawn in the plane with no $r$ pairwise crossing edges. Let $s \geq 3$ be an integer and $r=2^s$. We prove that there is a constant $C$ such that every $r$-quasiplanar graph with $n \geq r$ vertices has at most $n\left(Cs^{-1}\log n\right)^{2s-4}$ edges. A graph whose vertices are continuous curves in the plane, two being connected by an edge if and only if they intersect, is called a string graph. We show that for every $\epsilon>0$, there exists $\delta>0$ such that every string graph with $n$ vertices, whose chromatic number is at least $n^{\epsilon}$ contains a clique of size at least $n^{\delta}$. A clique of this size or a coloring using fewer than $n^{\epsilon}$ colors can be found by a polynomial time algorithm in terms of the size of the geometric representation of the set of strings. In the process, we use, generalize, and strengthen previous results of Lee, Tomon, and others. All of our theorems are related to geometric variants of the following classical graph-theoretic problem of Erdos, Gallai, and Rogers. Given a $K_r$-free graph on $n$ vertices and an integer $s

The size Ramsey number of a graph $H$ is defined as the minimum number of edges in a graph $G$ such that there is a monochromatic copy of $H$ in every two-coloring of $E(G)$. The size Ramsey number was introduced by Erd\H{o}s, Faudree, Rousseau, and Schelp in 1978 and they ended their foundational paper by asking whether one can determine up to a constant factor the size Ramsey numbers of three families of graphs: complete bipartite graphs, book graphs (obtained by adding many common neighbors to the vertices of a clique), and starburst graphs (obtained by adding many pendant edges to each vertex of a clique). In this paper, we completely resolve the latter two questions and make substantial progress on the first by determining the size Ramsey number of $K_{s,t}$ up to a constant factor for all $t = \Omega(s\log s)$., Comment: 23 pages

We prove that the the discrepancy of arithmetic progressions in the $d$-dimensional grid $\{1, \dots, N\}^d$ is within a constant factor depending only on $d$ of $N^{\frac{d}{2d+2}}$. This extends the case $d=1$, which is a celebrated result of Roth and of Matou\v{s}ek and Spencer, and removes the polylogarithmic factor from the previous upper bound of Valk\'o from about two decades ago. We further prove similarly tight bounds for grids of differing side lengths in many cases., Comment: 25 pages

The book graph $B_n^{(k)}$ consists of $n$ copies of $K_{k+1}$ joined along a common $K_k$. In the prequel to this paper, we studied the diagonal Ramsey number $r(B_n^{(k)}, B_n^{(k)})$. Here we consider the natural off-diagonal variant $r(B_{cn}^{(k)}, B_n^{(k)})$ for fixed $c \in (0,1]$. In this more general setting, we show that an interesting dichotomy emerges: for very small $c$, a simple $k$-partite construction dictates the Ramsey function and all nearly-extremal colorings are close to being $k$-partite, while, for $c$ bounded away from $0$, random colorings of an appropriate density are asymptotically optimal and all nearly-extremal colorings are quasirandom. Our investigations also open up a range of questions about what happens for intermediate values of $c$., Comment: 36 pages

The Ramsey number $r(G,H)$ is the minimum $N$ such that every graph on $N$ vertices contains $G$ as a subgraph or its complement contains $H$ as a subgraph. For integers $n \geq k \geq 1$, the $k$-book $B_{k,n}$ is the graph on $n$ vertices consisting of a copy of $K_k$, called the spine, as well as $n-k$ additional vertices each adjacent to every vertex of the spine and non-adjacent to each other. A connected graph $H$ on $n$ vertices is called $p$-good if $r(K_p,H)=(p-1)(n-1)+1$. Nikiforov and Rousseau proved that if $n$ is sufficiently large in terms of $p$ and $k$, then $B_{k,n}$ is $p$-good. Their proof uses Szemer\'edi's regularity lemma and gives a tower-type bound on $n$. We give a short new proof that avoids using the regularity method and shows that every $B_{k,n}$ with $n \geq 2^{k^{10p}}$ is $p$-good. Using Szemer\'edi's regularity lemma, Nikiforov and Rousseau also proved much more general goodness-type results, proving a tight bound on $r(G,H)$ for several families of sparse graphs $G$ and $H$ as long as $|V(G)| < \delta |V(H)|$ for a small constant $\delta > 0$. Using our techniques, we prove a new result of this type, showing that $r(G,H) = (p-1)(n-1)+1$ when $H =B_{k,n}$ and $G$ is a complete $p$-partite graph whose first $p-1$ parts have constant size and whose last part has size $\delta n$, for some small constant $\delta>0$. Again, our proof does not use the regularity method, and thus yields double-exponential bounds on $\delta$., Comment: 21 pages

We prove that the number of edges of a multigraph $G$ with $n$ vertices is at most $O(n^2\log n)$, provided that any two edges cross at most once, parallel edges are noncrossing, and the lens enclosed by every pair of parallel edges in $G$ contains at least one vertex. As a consequence, we prove the following extension of the Crossing Lemma of Ajtai, Chv\'atal, Newborn, Szemer\'edi and Leighton, if $G$ has $e \geq 4n$ edges, in any drawing of $G$ with the above property, the number of crossings is $\Omega\left(\frac{e^3}{n^2\log(e/n)}\right)$. This answers a question of Kaufmann et al. and is tight up to the logarithmic factor., Comment: Appears in the Proceedings of the 29th International Symposium on Graph Drawing and Network Visualization (GD 2021)

The Ramsey number $r(H)$ of a graph $H$ is the minimum integer $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. While this definition only asks for a single monochromatic copy of $H$, it is often the case that every two-edge-coloring of the complete graph on $r(H)$ vertices contains many monochromatic copies of $H$. The minimum number of such copies over all two-colorings of $K_{r(H)}$ will be referred to as the threshold Ramsey multiplicity of $H$. Addressing a problem of Harary and Prins, who were the first to systematically study this quantity, we show that there is a positive constant $c$ such that the threshold Ramsey multiplicity of a path or an even cycle on $k$ vertices is at least $(ck)^k$. This bound is tight up to the constant $c$. We prove a similar result for odd cycles in a companion paper., Comment: 28 pages

The Ramsey number $r(H)$ of a graph $H$ is the minimum $n$ such that any two-coloring of the edges of the complete graph $K_n$ contains a monochromatic copy of $H$. The threshold Ramsey multiplicity $m(H)$ is then the minimum number of monochromatic copies of $H$ taken over all two-edge-colorings of $K_{r(H)}$. The study of this concept was first proposed by Harary and Prins almost fifty years ago. In a companion paper, the authors have shown that there is a positive constant $c$ such that the threshold Ramsey multiplicity for a path or even cycle with $k$ vertices is at least $(ck)^k$, which is tight up to the value of $c$. Here, using different methods, we show that the same result also holds for odd cycles with $k$ vertices., Comment: 17 pages

The classical Erd\H{o}s-Littlewood-Offord theorem says that for nonzero vectors $a_1,\dots,a_n\in \mathbb{R}^d$, any $x\in \mathbb{R}^d$, and uniformly random $(\xi_1,\dots,\xi_n)\in\{-1,1\}^n$, we have $\Pr(a_1\xi_1+\dots+a_n\xi_n=x)=O(n^{-1/2})$. In this paper we show that $\Pr(a_1\xi_1+\dots+a_n\xi_n\in S)\le n^{-1/2+o(1)}$ whenever $S$ is definable with respect to an o-minimal structure (for example, this holds when $S$ is any algebraic hypersurface), under the necessary condition that it does not contain a line segment. We also obtain an inverse theorem in this setting., Comment: 22 pages, minor edits. To appear in the Annals of Probability

For which graphs $F$ is there a sparse $F$-counting lemma in $C_4$-free graphs? We are interested in identifying graphs $F$ with the property that, roughly speaking, if $G$ is an $n$-vertex $C_4$-free graph with on the order of $n^{3/2}$ edges, then the density of $F$ in $G$, after a suitable normalization, is approximately at least the density of $F$ in an $\epsilon$-regular approximation of $G$. In recent work, motivated by applications in extremal and additive combinatorics, we showed that $C_5$ has this property. Here we construct a family of graphs with the property., Comment: 13 pages

We show that in any two-coloring of the positive integers there is a color for which the set of positive integers that can be represented as a sum of distinct elements with this color has upper logarithmic density at least $(2+\sqrt{3})/4$ and this is best possible. This answers a forty-year-old question of Erd\H{o}s., Comment: 9 pages

The clique removal lemma says that for every $r \geq 3$ and $\varepsilon>0$, there exists some $\delta>0$ so that every $n$-vertex graph $G$ with fewer than $\delta n^r$ copies of $K_r$ can be made $K_r$-free by removing at most $\varepsilon n^2$ edges. The dependence of $\delta$ on $\varepsilon$ in this result is notoriously difficult to determine: it is known that $\delta^{-1}$ must be at least super-polynomial in $\varepsilon^{-1}$, and that it is at most of tower type in $\log \varepsilon^{-1}$. We prove that if one imposes an appropriate minimum degree condition on $G$, then one can actually take $\delta$ to be a linear function of $\varepsilon$ in the clique removal lemma. Moreover, we determine the threshold for such a minimum degree requirement, showing that above this threshold we have linear bounds, whereas below the threshold the bounds are once again super-polynomial, as in the unrestricted removal lemma. We also investigate this question for other graphs besides cliques, and prove some general results about how minimum degree conditions affect the bounds in the graph removal lemma., Comment: 18 pages

Burr and Erd\H{o}s in 1975 conjectured, and Chv\'atal, R\"odl, Szemer\'edi and Trotter later proved, that the Ramsey number of any bounded degree graph is linear in the number of vertices. In this paper, we disprove the natural directed analogue of the Burr--Erd\H{o}s conjecture, answering a question of Buci\'c, Letzter, and Sudakov. If $H$ is an acyclic digraph, the oriented Ramsey number of $H$, denoted $\overrightarrow{r_{1}}(H)$, is the least $N$ such that every tournament on $N$ vertices contains a copy of $H$. We show that for any $\Delta \geq 2$ and any sufficiently large $n$, there exists an acyclic digraph $H$ with $n$ vertices and maximum degree $\Delta$ such that \[ \overrightarrow{r_{1}}(H)\ge n^{\Omega(\Delta^{2/3}/ \log^{5/3} \Delta)}. \] This proves that $\overrightarrow{r_{1}}(H)$ is not always linear in the number of vertices for bounded-degree $H$. On the other hand, we show that $\overrightarrow{r_{1}}(H)$ is nearly linear in the number of vertices for typical bounded-degree acyclic digraphs $H$, and obtain linear or nearly linear bounds for several natural families of bounded-degree acyclic digraphs. For multiple colors, we prove a quasi-polynomial upper bound $\overrightarrow{r_{k}}(H)=2^{(\log n)^{O_{k}(1)}}$ for all bounded-degree acyclic digraphs $H$ on $n$ vertices, where $\overrightarrow{r_k}(H)$ is the least $N$ such that every $k$-edge-colored tournament on $N$ vertices contains a monochromatic copy of $H$. For $k\geq 2$ and $n\geq 4$, we exhibit an acyclic digraph $H$ with $n$ vertices and maximum degree $3$ such that $\overrightarrow{r_{k}}(H)\ge n^{\Omega(\log n/\log\log n)}$, showing that these Ramsey numbers can grow faster than any polynomial in the number of vertices., Comment: 28 pages; revised to reflect referee comments

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 Erd\H{o}s on Ramsey complete sequences, for which Erd\H{o}s 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 Erd\H{o}s on the minimum number of colors needed to color the positive integers less than $n$ so that $n$ cannot be written as a monochromatic sum; the exact determination of an extremal function introduced by Erd\H{o}s and Graham on sets of integers avoiding a given subset sum; and, answering a question reiterated by several authors, a homogeneous strengthening of a seminal result of Szemer\'edi and Vu on long arithmetic progressions in subset sums., Comment: 75 pages

We study quantitative relationships between the triangle removal lemma and several of its variants. One such variant, which we call the triangle-free lemma, states that for each $\epsilon>0$ there exists $M$ such that every triangle-free graph $G$ has an $\epsilon$-approximate homomorphism to a triangle-free graph $F$ on at most $M$ vertices (here an $\epsilon$-approximate homomorphism is a map $V(G) \to V(F)$ where all but at most $\epsilon |V(G)|^2$ edges of $G$ are mapped to edges of $F$). One consequence of our results is that the least possible $M$ in the triangle-free lemma grows faster than exponential in any polynomial in $\epsilon^{-1}$. We also prove more general results for arbitrary graphs, as well as arithmetic analogues over finite fields, where the bounds are close to optimal., Comment: 16 pages, 2 figures

Celebrated theorems of Roth and of Matou\v{s}ek and Spencer together show that the discrepancy of arithmetic progressions in the first $n$ positive integers is $\Theta(n^{1/4})$. We study the analogous problem in the $\mathbb{Z}_n$ setting. We asymptotically determine the logarithm of the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ for all positive integer $n$. We further determine up to a constant factor the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ for many $n$. For example, if $n=p^k$ is a prime power, then the discrepancy of arithmetic progressions in $\mathbb{Z}_n$ is $\Theta(n^{1/3+r_k/(6k)})$, where $r_k \in \{0,1,2\}$ is the remainder when $k$ is divided by $3$. This solves a problem of Hebbinghaus and Srivastav., Comment: 22 pages + 4 pages appendix

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., Comment: 7 pages

Given a family $\mathcal F$ of $k$-element sets, $S_1,\ldots,S_r\in\mathcal F$ form an {\em $r$-sunflower} if $S_i \cap S_j =S_{i'} \cap S_{j'}$ for all $i \neq j$ and $i' \neq j'$. According to a famous conjecture of Erd\H os and Rado (1960), there is a constant $c=c(r)$ such that if $|\mathcal F|\ge c^k$, then $\mathcal F$ contains an $r$-sunflower. We come close to proving this conjecture for families of bounded {\em Vapnik-Chervonenkis dimension}, VC-dim$(\mathcal F)\le d$. In this case, we show that $r$-sunflowers exist under the slightly stronger assumption $|\mathcal F|\ge2^{10k(dr)^{2\log^{*} k}}$. Here, $\log^*$ denotes the iterated logarithm function. We also verify the Erd\H os-Rado conjecture for families $\mathcal F$ of bounded {\em Littlestone dimension} and for some geometrically defined set systems.

We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases of this question. For $H$ any fixed odd cycle, we determine the answer up to a constant factor when $n$ is sufficiently large. We also prove an upper bound when $H$ is a fixed clique that we conjecture is tight up to a constant factor, and prove upper bounds for more general families of graphs. We apply our results to get a new bound on the maximum cut of graphs with a forbidden odd cycle in terms of the number of edges., Comment: 21 pages