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In 1972, Branko Gr\"unbaum conjectured that any arrangement of $n>2$ pairwise crossing pseudocircles in the plane can have at most $2n-2$ digons (regions enclosed by exactly two pseudoarcs), with the bound being tight. While this conjecture has been confirmed for cylindrical arrangements of pseudocircles and more recently for geometric circles, we extend these results to any simple arrangement of pairwise intersecting pseudocircles. Using techniques from the above-mentioned special cases, we provide a complete proof of Gr\"unbaum's conjecture that has stood open for over five decades.

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., Comment: 10 pages

A long-standing open conjecture of Branko Gr\"unbaum from 1972 states that any simple arrangement of $n$ pairwise intersecting pseudocircles in the plane can have at most $2n-2$ digons. Agarwal et al. proved this conjecture for arrangements of pairwise intersecting pseudocircles in which there is a common point surrounded by all pseudocircles. Recently, Felsner, Roch and Scheucher showed that Gr\"unbaum's conjecture is true for arrangements of pairwise intersecting pseudocircles in which there are three pseudocircles every pair of which create a digon. In this paper we prove this over 50-year-old conjecture of Gr\"unbaum for any simple arrangement of pairwise intersecting circles in the plane.

In this paper, we consider saturation problems related to the celebrated Erd\H{o}s--Szekeres convex polygon problem. For each $n \ge 7$, we construct a planar point set of size $(7/8) \cdot 2^{n-2}$ which is saturated for convex $n$-gons. That is, the set contains no $n$ points in convex position while the addition of any new point creates such a configuration. This demonstrates that the saturation number is smaller than the Ramsey number for the Erd\H{o}s--Szekeres problem. The proof also shows that the original Erd\H{o}s--Szekeres construction is indeed saturated. Our construction is based on a similar improvement for the saturation version of the cups-versus-caps theorem. Moreover, we consider the generalization of the cups-versus-caps theorem to monotone paths in ordered hypergraphs. In contrast to the geometric setting, we show that this abstract saturation number is always equal to the corresponding Ramsey number., Comment: Minor revisions

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.

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.

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 Erd\H{o}s-Sands-Sauer-Woodrow conjecture. Surprisingly, the proof also relies on the two dimensional case of the Illumination conjecture.

Thomassen formulated the following conjecture: Every $3$-connected cubic graph has a red-blue vertex coloring such that the blue subgraph has maximum degree at most $1$ (that is, it consists of a matching and some isolated vertices) and the red subgraph has minimum degree at least $1$ and contains no $3$-edge path. Since all monochromatic components are small in this coloring and there is a certain irregularity, we call such a coloring \emph{crumby}. Recently, Bellitto, Klimo\v{s}ov\'a, Merker, Witkowski and Yuditsky \cite{counter} constructed an infinite family refuting the above conjecture. Their prototype counterexample is $2$-connected, planar, but contains a $K_4$-minor and also a $5$-cycle. This leaves the above conjecture open for some important graph classes: outerplanar graphs, $K_4$-minor-free graphs, bipartite graphs. In this regard, we prove that $2$-connected outerplanar graphs, subdivisions of $K_4$ and $1$-subdivisions of cubic graphs admit crumby colorings. A subdivision of $G$ is {\it genuine} if every edge is subdivided at least once. We show that every genuine subdivision of any subcubic graph admits a crumby coloring. We slightly generalise some of these results and formulate a few conjectures., Comment: 22 pages, 15 figures, revised version

We prove that for every $m$ there is a finite point set $\mathcal{P}$ in the plane such that no matter how $\mathcal{P}$ is three-colored, there is always a disk containing exactly $m$ points, all of the same color. This improves a result of Pach, Tardos and T\'oth who proved the same for two colors. The main ingredient of the construction is a subconstruction whose points are in convex position. Namely, we show that for every $m$ there is a finite point set $\mathcal{P}$ in the plane in convex position such that no matter how $\mathcal{P}$ is two-colored, there is always a disk containing exactly $m$ points, all of the same color. We also prove that for unit disks no similar construction can work, and several other results., Comment: 17 pages, 7 figures

An odd wheel graph is a graph formed by connecting a new vertex to all vertices of an odd cycle. We answer a question of Rosenfeld and Le by showing that odd wheels cannot be drawn in the plane such that the lengths of the edges are odd integers., Comment: 11 pages, 3 figures. Appears in the Proceedings of the 28th International Symposium on Graph Drawing and Network Visualization (GD 2020)

In 1964, Erd\H{o}s, Hajnal and Moon introduced a saturation version of Tur\'an's classical theorem in extremal graph theory. In particular, they determined the minimum number of edges in a $K_r$-free, $n$-vertex graph with the property that the addition of any further edge yields a copy of $K_r$. We consider analogues of this problem in other settings. We prove a saturation version of the Erd\H{o}s-Szekeres theorem about monotone subsequences and saturation versions of some Ramsey-type theorems on graphs and Dilworth-type theorems on posets. We also consider semisaturation problems, wherein we allow the family to have the forbidden configuration, but insist that any addition to the family yields a new copy of the forbidden configuration. In this setting, we prove a semisaturation version of the Erd\H{o}s-Szekeres theorem on convex $k$-gons, as well as multiple semisaturation theorems for sequences and posets.

We study covering numbers and local covering numbers with respect to difference graphs and complete bipartite graphs. In particular we show that in every cover of a Young diagram with $\binom{2k}{k}$ steps with generalized rectangles there is a row or a column in the diagram that is used by at least $k+1$ rectangles, and prove that this is best-possible. This answers two questions by Kim, Martin, Masa{\v{r}}{\'\i}k, Shull, Smith, Uzzell, and Wang (Europ. J. Comb. 2020), namely: - What is the local complete bipartite cover number of a difference graph? - Is there a sequence of graphs with constant local difference graph cover number and unbounded local complete bipartite cover number? We add to the study of these local covering numbers with a lower bound construction and some examples. Following Kim \emph{et al.}, we use the results on local covering numbers to provide lower and upper bounds for the local dimension of partially ordered sets of height~2. We discuss the local dimension of some posets related to Boolean lattices and show that the poset induced by the first two layers of the Boolean lattice has local dimension $(1 + o(1))\log_2\log_2 n$. We conclude with some remarks on covering numbers for digraphs and Ferrers dimension., Comment: Parts of this paper have previously been reported in arXiv submission arXiv:1902.08223

We prove the following Helly-type result. Let $\mathcal{C}_1,\dots,\mathcal{C}_{3d}$ be finite families of convex bodies in $\mathbb{R}^d$. Assume that for any colorful selection of $2d$ sets, $C_{i_k}\in \mathcal{C}_{i_k}$ for each $1\leq k\leq 2d$ with $1\leq i_1<\dots

Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists) if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by $m(G)$. It was shown by Saks and Werman that $m(K_n)=n-b(n)$, where $b(n)$ is the number of 1's in the binary representation of $n$. In this paper, we initiate the study of the problem for general graphs. The obvious bounds for a connected graph $G$ on $n$ vertices are $n-b(n)\le m(G)\le n-1$. We show that for any tree $T$ on an even number of vertices we have $m(T)=n-1$ and that for any tree $T$ on an odd number of vertices, we have $n-65\le m(T)\le n-2$. Our proof uses results about the weighted version of the problem for $K_n$, which may be of independent interest. We also exhibit a sequence $G_n$ of graphs with $m(G_n)=n-b(n)$ such that $G_n$ has $O(nb(n))$ edges and $n$ vertices., Comment: 19 pages

A widely investigated subject in combinatorial geometry, originated from Erd\H{o}s, is the following. Given a point set $P$ of cardinality $n$ in the plane, how can we describe the distribution of the determined distances? This has been generalized in many directions. In this paper we propose the following variants. Consider planar arrangements of $n$ lines. Determine the maximum number of triangles of unit area, maximum area or minimum area, determined by these lines. Determine the minimum size of a subset of these $n$ lines so that all triples determine distinct area triangles. We prove that the order of magnitude for the maximum occurrence of unit areas lies between $\Omega(n^2)$ and $O(n^{9/4})$. This result is strongly connected to both additive combinatorial results and Szemer\'edi--Trotter type incidence theorems. Next we show a tight bound for the maximum number of minimum area triangles. Finally we present lower and upper bounds for the maximum area and distinct area problems by combining algebraic, geometric and combinatorial techniques., Comment: Title is shortened. Some typos and small errors were corrected