A cyclic non-Ramsey heptagon
We present a construction with seven points on a circle that is not Ramsey, disproving a conjecture of Erdős, Graham, Montgomery, Rothschild, Spencer and Straus.
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arXiv subjects
Dömötör Pálvölgyi
Publications and source records attributed to Dömötör Pálvölgyi.
A nearcircumsphere-Ramsey Theorem for Solvable Transitive Configurations
Let P be a finite spherical set. We prove that if P admits a solvable group of isometries acting transitively on it, then every r-coloring of a sufficiently high-dimensional sphere of radius slightly larger than the circumradius of P contains a monochromatic congruent copy of P. Our proof builds on the group-theoretic argument of Kriz and combines it with a topological method that may be of independent interest.
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Note on polychromatic coloring of hereditary hypergraph families II
We extend a recent construction concerning polychromatic colorings of hereditary hypergraph families. For every integer $h\ge 4$ we construct a $(2h-1)$-uniform hypergraph which has no polychromatic $3$-coloring, but all of whose $h$-heavy restricted subhypergraphs are $2$-colorable. Together with the previously known case $h=3$, this gives examples with uniformity $2h-1$ for every $h\ge 3$. The construction is based on complements of suitable $h$-uniform hypergraphs on $3h-1$ vertices. For $h\ge 9$ we prove existence by a simple probabilistic argument; the remaining cases $4\le h\le 8$ are certified by a short exhaustive computer check, whose fully reproducible description and source code are included in the appendix.
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Unit distance graphs with few crossings per edge
A graph is called a $k$-planar unit distance graph if it can be drawn in the plane such that every edge is a unit line segment and is involved in at most $k$ crossings. We investigate $u_k(n)$, the maximum number of edges of such graphs on $n$ vertices. For $k=1$, we improve the best known upper bound, by showing that $u_1(n) \leq 3n - c\sqrt{n}$ for some constant $c>0$. This bound is tight up to the value of the constant $c$. For $k=2$, we establish the first non-trivial upper bound by proving that $u_2(n) \leq 4n - 8$. Regarding lower bounds we give a construction for $k=2$ that shows $u_2(n) \geq u_0(n) + c\sqrt{n}$ if $n$ is sufficiently large.
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Erdős-Szekeres Maker-Breaker Games
We present new results on Maker-Breaker games arising from the Erdős-Szekeres problem in planar geometry. This classical problem asks how large a set in general position has to be to ensure the existence of $n$ points that are the vertices of a convex $n$-gon. Moreover, Erdős further extended this problem by asking what happens if we also require that this $n$-gon has an empty interior. In a 2-player Maker-Breaker setting, this problem inspires two main games. In both games, Maker tries to obtain an empty convex $k$-gon, while Breaker tries to prevent her from doing so. The games differ only in which points can comprise the winning $k$-gons: in the monochromatic version the points of both players can make up a $k$-gon, while in the bichromatic version only Maker's points contribute to such a polygon. Both settings are studied in this paper. We show that in the monochromatic game, Maker always wins. Even in a biased game where Breaker is allowed to place $s$ points per round, for any constant $s \geq 1$, Maker has a winning strategy. In the bichromatic setting, Maker still wins whenever Breaker is allowed to place $s$ points per round for any constant $s<2$. This settles an open problem posed by Aichholzer et al. (2019). Furthermore, we show that there are games that are not a lost cause for Breaker. Whenever $k\ge 8$ and Breaker is allowed to play 12 or more points per round, she has a winning strategy. We also consider the one-round bichromatic game (a.k.a.\ the offline version). In this setting, we show that Breaker wins if she can place twice as many points as Maker but if the bias is less than $2$, then Maker wins for large enough set of points.
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Three-chromatic geometric hypergraphs
We prove that for any planar convex body C there is a positive integer m with the property that any finite point set P in the plane can be three-colored such that there is no translate of C containing at least m points of P, all of the same color. As a part of the proof, we show a strengthening of the Erdős-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.
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Coloring Geometric Hypergraphs: A Survey
The \emph{chromatic number} of a hypergraph is the smallest number of colors needed to color the vertices such that no edge of at least two vertices is monochromatic. Given a family of geometric objects $\mathcal{F}$ that covers a subset $S$ of the Euclidean space, we can associate it with a hypergraph whose vertex set is $\mathcal F$ and whose edges are those subsets ${\mathcal{F}'}\subset \mathcal F$ for which there exists a point $p\in S$ such that ${\mathcal F}'$ consists of precisely those elements of $\mathcal{F}$ that contain $p$. The question whether $\mathcal F$ can be split into 2 coverings is equivalent to asking whether the chromatic number of the hypergraph is equal to 2. There are a number of competing notions of the chromatic number that lead to deep combinatorial questions already for abstract hypergraphs. In this paper, we concentrate on \emph{geometrically defined} (in short, \emph{geometric}) hypergraphs, and survey many recent coloring results related to them. In particular, we study and survey the following problem, dual to the above covering question. Given a set of points $S$ in the Euclidean space and a family $\mathcal{F}$ of geometric objects of a fixed type, define a hypergraph ${\mathcal H}_m$ on the point set $S$, whose edges are the subsets of $S$ that can be obtained as the intersection of $S$ with a member of $\mathcal F$ and have at least $m$ elements. Is it true that if $m$ is large enough, then the chromatic number of ${\mathcal H}_m$ is equal to 2?
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Orientation of good covers
We study systems of orientations on triples that satisfy the following so-called interiority condition: $\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ implies $\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O; we denote a P3O realizable by points as p-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-T3O, is the order type of a point set in general position. In our paper "Orientation of convex sets" we defined a 3-order on pairwise intersecting convex sets; such a P3O is called a C-P3O. In this paper we extend this 3-order to pairwise intersecting good covers; such a P3O is called a GC-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a C-T3O and a GC-T3O, respectively. The main result of this paper is that there is a p-T3O that is not a GC-T3O, implying also that it is not a C-T3O -- this latter problem was left open in our earlier paper. Our proof involves several combinatorial and geometric observations that can be of independent interest. Along the way, we define several further special families of GC-T3O's.
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Stabbing non-piercing sets and face lengths in large girth plane graphs
We show that a non-piercing family of connected planar sets with bounded independence number can be stabbed with a constant number of points. As a consequence, we answer a question of Axenovich, Kießle and Sagdeev about the largest possible face length of an edge-maximal plane graph with girth at least $\ell$.
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Monochromatic configurations on a circle
If we two-colour a circle, we can always find an inscribed triangle with angles $(\fracπ{7},\frac{2π}{7},\frac{4π}{7})$ whose three vertices have the same colour. In fact, Bialostocki and Nielsen showed that it is enough to consider the colours on the vertices of an inscribed heptagon. We prove that for every other triangle $T$ there is a two-colouring of the circle without any monochromatic copy of $T$. More generally, for $k\geq 3$, call a $k$-tuple $(d_1,d_2,\dots,d_k)$ with $d_1\geq d_2\geq \dots \geq d_k>0$ and $\sum_{i=1}^k d_i=1$ a Ramsey $k$-tuple if the following is true: in every two-colouring of the circle of unit perimeter, there is a monochromatic $k$-tuple of points in which the distances of cyclically consecutive points, measured along the arcs, are $d_1,d_2,\dots,d_k$ in some order. By a conjecture of Stromquist, if $d_i=\frac{2^{k-i}}{2^k-1}$, then $(d_1,\dots,d_k)$ is Ramsey. Our main result is a proof of the converse of this conjecture. That is, we show that if $(d_1,\dots,d_k)$ is Ramsey, then $d_i=\frac{2^{k-i}}{2^k-1}$. We do this by finding connections of the problem to certain questions from number theory about partitioning $\mathbb{N}$ into so-called Beatty sequences. We also disprove a majority version of Stromquist's conjecture, study a robust version, and discuss a discrete version.
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The complexity of recognizing $ABAB$-free hypergraphs
The study of geometric hypergraphs gave rise to the notion of $ABAB$-free hypergraphs. A hypergraph $\mathcal{H}$ is called $ABAB$-free if there is an ordering of its vertices such that there are no hyperedges $A,B$ and vertices $v_1,v_2,v_3,v_4$ in this order satisfying $v_1,v_3\in A\setminus B$ and $v_2,v_4\in B\setminus A$. In this paper, we prove that it is NP-complete to decide if a hypergraph is $ABAB$-free. We show a number of analogous results for hypergraphs with similar forbidden patterns, such as $ABABA$-free hypergraphs. As an application, we show that deciding whether a hypergraph is realizable as the incidence hypergraph of points and pseudodisks is also NP-complete.
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Piercing intersecting convex sets
Assume two finite families $\mathcal A$ and $\mathcal B$ of convex sets in $\mathbb{R}^3$ have the property that $A\cap B\ne \emptyset$ for every $A \in \mathcal A$ and $B\in \mathcal B$. Is there a constant $γ>0$ (independent of $\mathcal A$ and $\mathcal B$) such that there is a line intersecting $γ|\mathcal A|$ sets in $\mathcal A$ or $γ|\mathcal B|$ sets in $\mathcal B$? This is an intriguing Helly-type question from a paper by Martínez, Roldan and Rubin. We confirm this in the special case when all sets in $\mathcal A$ lie in parallel planes and all sets in $\mathcal B$ lie in parallel planes; in fact, all sets from one of the two families has a line transversal.
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$k$-dimensional transversals for fat convex sets
We prove a fractional Helly theorem for $k$-flats intersecting fat convex sets. A family $\mathcal{F}$ of sets is said to be $ρ$-fat if every set in the family contains a ball and is contained in a ball such that the ratio of the radii of these balls is bounded by $ρ$. We prove that for every dimension $d$ and positive reals $ρ$ and $α$ there exists a positive $β=β(d,ρ, α)$ such that if $\mathcal{F}$ is a finite family of $ρ$-fat convex sets in $\mathbb{R}^d$ and an $α$-fraction of the $(k+2)$-size subfamilies from $\mathcal{F}$ can be hit by a $k$-flat, then there is a $k$-flat that intersects at least a $β$-fraction of the sets of $\mathcal{F}$. We prove spherical and colorful variants of the above results and prove a $(p,k+2)$-theorem for $k$-flats intersecting balls.
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A note on infinite versions of $(p,q)$-theorems
We prove that fractional Helly and $(p,q)$-theorems imply $(\aleph_0,q)$-theorems in an entirely abstract setting. We give a plethora of applications, including reproving almost all earlier $(\aleph_0,q)$-theorems about geometric hypergraphs that were proved recently. Some of the corollaries are new results, for example, we prove that if $\mathcal{F}$ is an infinite family of convex compact sets in $\mathbb{R}^d$ and among every $\aleph_0$ of the sets some $d+1$ contain a point in their intersection with integer coordinates, then all the members of $\mathcal{F}$ can be hit with finitely many points with integer coordinates.
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On dual-ABAB-free and related hypergraphs
Geometric motivations warranted the study of hypergraphs on ordered vertices that have no pair of hyperedges that induce an alternation of some given length. Such hypergraphs are called ABA-free, ABAB-free and so on. Since then various coloring and other combinatorial results were proved about these families of hypergraphs. We prove a characterization in terms of their incidence matrices which avoids using the ordering of the vertices. Using this characterization, we prove new results about the dual hypergraphs of ABAB-free hypergraphs. In particular, we show that dual-ABAB-free hypergraphs are not always proper $2$-colorable even if we restrict ourselves to hyperedges that are larger than some parameter $m$.
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Query complexity of Boolean functions on the middle slice of the cube
We study the query complexity on slices of Boolean functions. Among other results we show that there exists a Boolean function for which we need to query all but 7 input bits to compute its value, even if we know beforehand that the number of 0's and 1's in the input are the same, i.e., when our input is from the middle slice. This answers a question of Byramji. Our proof is non-constructive, but we also propose a concrete candidate function that might have the above property. Our results are related to certain natural discrepancy type questions that, somewhat surprisingly, have not been studied before.
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The double Hall property and cycle covers in bipartite graphs
In a graph $G$, the $2$-neighborhood of a vertex set $X$ consists of all vertices of $G$ having at least $2$ neighbors in $X$. We say that a bipartite graph $G(A,B)$ satisfies the double Hall property if $|A|\geq2$, and every subset $X \subseteq A$ of size at least $2$ has a $2$-neighborhood of size at least $|X|$. Salia conjectured that any bipartite graph $G(A,B)$ satisfying the double Hall property contains a cycle covering $A$. Here, we prove the existence of a $2$-factor covering $A$ in any bipartite graph $G(A,B)$ satisfying the double Hall property. We also show Salia's conjecture for graphs with restricted degrees of vertices in $B$. Additionally, we prove a lower bound on the number of edges in a graph satisfying the double Hall property, and the bound is sharp up to a constant factor.
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Orientation of convex sets
We introduce a novel definition of orientation on the triples of a family of pairwise intersecting planar convex sets and study its properties. In particular, we compare it to other systems of orientations on triples that satisfy a so-called interiority condition: $\circlearrowleft(ABD)=~\circlearrowleft(BCD)=~\circlearrowleft(CAD)=1$ imply $\circlearrowleft(ABC)=1$ for any $A,B,C,D$. We call such an orientation a P3O (partial 3-order), a natural generalization of a poset, that has several interesting special cases. For example, the order type of a planar point set (that can have collinear triples) is a P3O; we denote a P3O realizable by points as p-P3O. If we do not allow $\circlearrowleft(ABC)=0$, we obtain a T3O (total 3-order). Contrary to linear orders, a T3O can have a rich structure. A T3O realizable by points, a p-T3O, is the order type of a point set in general position. Despite these similarities to order types, P3O and p-T3O that can arise from the orientation of pairwise intersecting convex sets, denoted by C-P3O and C-T3O, turn out to be quite different from order types: there is no containment relation among the family of all C-P3O's and the family of all p-P3O's, or among the families of C-T3O's and p-T3O's. Finally, we study properties of these orientations if we also require that the family of underlying convex sets satisfies the (4,3) property.
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