arXiv subjects
Publications and source records attributed to Janos Pach.
Let $k\ge 2$ be fixed integer, $0<c<1$ a constant. Consider a graph $G$ with $n$ vertices and average degree $cn$. We answer a question of Simon Griffiths by showing that $G$ has $k$ vertices such that their neighborhoods together cover at least $\min(1-(1-c)^{k},\sqrt{c})n$ vertices. This result is essentially tight.
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 $2^{Ω(n^{4/3})}$ 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 $2^{O(n^{4/3}\log^2n)}$. 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.
Raimi's theorem guarantees the existence of a partition of $\mathbb{N}$ into two parts with an unavoidable intersection property: for any finite coloring of $\mathbb{N}$, some color class intersects both parts infinitely many times, after an appropriate shift (translation). We establish a polynomial extension of this result, proving that such intersections persist under polynomial shifts in any dimension. Let $P^{(1)},\dots,P^{(f)}\in\mathbb{Z}[x]$ be non-constant polynomials with positive leading coefficients and $P^{(j)}(0)=0$ for every $j$. We construct a partition of $\mathbb{N}^k$ into an arbitrarily fixed finite number of pieces such that for any coloring of $\mathbb{N}^k$ with finitely many colors, there exist $x_0\in \mathbb{N}$ and a single color class that meets all partition pieces after shifts by $x_0+P^{(j)}(h)$ in each of the $k$ coordinate directions, for every $j$ and infinitely many values $h\in \mathbb{N}$. Our proof exploits Weyl's equidistribution theory, Pontryagin duality, and the structure of polynomial relation lattices. We also prove some finite analogues of the above results for abelian groups and $SL_2(\mathbb{F}_q)$.
A plank is the part of space between two parallel planes. The following open problem, posed 45 years ago, can be viwed as the converse of Tarski's plank problem (Bang's theorem): Is it true that if the total width of a collection of planks is sufficiently large, then the planks can be individually translated to cover a unit ball $B$? A translative covering of $B$ by planks is said to be non-dissective if the planks can be added one by one, in some order, such that the uncovered part remains connected at each step, and is empty at the end. Improving a classical result of Groemer, we show that every set of $C/ε^{7/4}$ planks of width $ε$ admits a non-dissective translative covering of $B$, provided $C$ is large enough. Our proof yields a low-complexity algorithm. We also establish the first nontrivial lower bound of $c/ε^{4/3}$ for this quantity.
A graph is said to contain $K_k$ (a clique of size $k$) as a weak immersion if it has $k$ vertices, pairwise connected by edge-disjoint paths. In 1989, Lescure and Meyniel made the following conjecture related to Hadwiger's conjecture: Every graph of chromatic number $k$ contains $K_k$ as a weak immersion. We prove this conjecture for graphs with at most $(1.64-o(1))k$ vertices. As an application, we make some progress on Albertson's conjecture, according to which every graph $G$ with chromatic number $k$ satisfies $cr(G) \geq cr(K_k)$. In particular, we show that the conjecture is true for all graphs of chromatic number $k$, provided that they have at most $(1.64-o(1))k$ vertices.
Let $p_0,\ldots,p_n$ be a finite sequence of points in an Euclidean space $\R^d$. Suppose that there is a (pointlike) particle sitting at each point $p_i$. In a ``legal'' move, any one of them can jump over another, landing on the other side, at exactly the same distance. Under what circumstances can we guarantee that for any $\varepsilon>0$ and any other sequence of points $q_0,\ldots, q_n\in\R^d$, there is a finite sequence of legal moves that takes the particle at $p_i$ to the $\varepsilon$-neighborhood of $q_i$, simultaneously for every $i$? We prove that this is possible if and only if the additive group generated by the vectors $p_1-p_0,\ldots,p_n-p_0$ is dense in $\R^d$.
We prove a far-reaching strengthening of Szemerédi'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.
A graph G = (V,E) is called fully regular if for every independent set $I\subset V$ , the number of vertices in $V\setminus$ I that are not connected to any element of I depends only on the size of I. A linear ordering of the vertices of G is called successive if for every i, the first i vertices induce a connected subgraph of G. We give an explicit formula for the number of successive vertex orderings of a fully regular graph. As an application of our results, we give alternative proofs of two theorems of Stanley and Gao + Peng, determining the number of linear edge orderings of complete graphs and complete bipartite graphs, respectively, with the property that the first i edges induce a connected subgraph. As another application, we give a simple product formula for the number of linear orderings of the hyperedges of a complete 3-partite 3-uniform hypergraph such that, for every i, the first i hyperedges induce a connected subgraph. We found similar formulas for complete (non-partite) 3-uniform hypergraphs and in another closely related case, but we managed to verify them only when the number of vertices is small.
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 $ε>0$, there exists $δ>0$ such that every string graph with $n$ vertices, whose chromatic number is at least $n^ε$ contains a clique of size at least $n^δ$. A clique of this size or a coloring using fewer than $n^ε$ 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 $s<r$, at least how many vertices can we find such that the subgraph induced by them is $K_s$-free?
Given a triangle $Δ$, we study the problem of determining the smallest enclosing and largest embedded isosceles triangles of $Δ$ with respect to area and perimeter. This problem was initially posed by Nandakumar and was first studied by Kiss, Pach, and Somlai, who showed that if $Δ'$ is the smallest area isosceles triangle containing $Δ$, then $Δ'$ and $Δ$ share a side and an angle. In the present paper, we prove that for any triangle $Δ$, every maximum area isosceles triangle embedded in $Δ$ and every maximum perimeter isosceles triangle embedded in $Δ$ shares a side and an angle with $Δ$. Somewhat surprisingly, the case of minimum perimeter enclosing triangles is different: there are infinite families of triangles $Δ$ whose minimum perimeter isosceles containers do not share a side and an angle with $Δ$.
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, Chvátal, Newborn, Szemerédi and Leighton, if $G$ has $e \geq 4n$ edges, in any drawing of $G$ with the above property, the number of crossings is $Ω\left(\frac{e^3}{n^2\log(e/n)}\right)$. This answers a question of Kaufmann et al. and is tight up to the logarithmic factor.
Given a polygonal region containing a target point (which we assume is the origin), it is not hard to see that there are two points on the perimeter that are antipodal, that is, whose midpoint is the origin. We prove three generalizations of this fact. (1) For any polygon (or any compact planar set) containing the origin, it is possible to place a given set of weights on the boundary so that their barycenter (center of mass) coincides with the origin, provided that the largest weight does not exceed the sum of the other weights. (2) On the boundary of any 3-dimensional compact set containing the origin, there exist three points that form an equilateral triangle centered at the origin. (3) For any $d$-dimensional bounded convex polyhedron containing the origin, there exists a pair of antipodal points consisting of a point on a $\lfloor d/2 \rfloor$-face and a point on a $\lceil d/2\rceil$-face.
Given $n$ pairwise disjoint sets $X_1,\ldots, X_n$, we call the elements of $S=X_1\times\ldots\times X_n$ strings. A nonempty set of strings $W\subseteq S$ is said to be well-connected if for every $v\in W$ and for every $i\, (1\le i\le n)$, there is another element $v'\in W$ which differs from $v$ only in its $i$th coordinate. We prove a conjecture of Yaokun Wu and Yanzhen Xiong by showing that every set of more than $\prod_{i=1}^n|X_i|-\prod_{i=1}^n(|X_i|-1)$ strings has a well-connected subset. This bound is tight.
The {\em disjointness graph} $G=G({\cal S})$ of a set of segments ${\cal S}$ in $R^d$, $d\ge 2,$ is a graph whose vertex set is ${\cal S}$ and two vertices are connected by an edge if and only if the corresponding segments are disjoint. We prove that the chromatic number of $G$ satisfies $χ(G)\le(ω(G))^4+(ω(G))^3$, where $ω(G)$ denotes the clique number of $G$. It follows, that $\cal S$ has $Ω(n^{1/5})$ pairwise intersecting or pairwise disjoint elements. Stronger bounds are established for lines in space, instead of segments. We show that computing $ω(G)$ and $χ(G)$ for disjointness graphs of lines in space are NP-hard tasks. However, we can design efficient algorithms to compute proper colorings of $G$ in which the number of colors satisfies the above upper bounds. One cannot expect similar results for sets of continuous arcs, instead of segments, even in the plane. We construct families of arcs whose disjointness graphs are triangle-free ($ω(G)=2$), but whose chromatic numbers are arbitrarily large.
Let $G$ be a multigraph with $n$ vertices and $e>4n$ edges, drawn in the plane such that any two parallel edges form a simple closed curve with at least one vertex in its interior and at least one vertex in its exterior. Pach and Tóth (A Crossing Lemma for Multigraphs, SoCG 2018) extended the Crossing Lemma of Ajtai et al. (Crossing-free subgraphs, North-Holland Mathematics Studies, 1982) and Leighton (Complexity issues in VLSI, Foundations of computing series, 1983) by showing that if no two adjacent edges cross and every pair of nonadjacent edges cross at most once, then the number of edge crossings in $G$ is at least $αe^3/n^2$, for a suitable constant $α>0$. The situation turns out to be quite different if nonparallel edges are allowed to cross any number of times. It is proved that in this case the number of crossings in $G$ is at least $αe^{2.5}/n^{1.5}$. The order of magnitude of this bound cannot be improved.
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.
Let $X$ be an $n$-element set, where $n$ is even. We refute a conjecture of J. Gordon and Y. Teplitskaya, according to which, for every maximal intersecting family $\mathcal{F}$ of $\frac{n}2$-element subsets of $X$, one can partition $X$ into $\frac{n}2$ disjoint pairs in such a way that no matter how we pick one element from each of the first $\frac{n}2 - 1$ pairs, the set formed by them can always be completed to a member of $\mathcal{F}$ by adding an element of the last pair. The above problem is related to classical questions in extremal set theory. For any $t\ge 2$, we call a family of sets $\mathcal{F}\subset 2^X$ {\em $t$-separable} if for any ordered pair of elements $(x,y)$ of $X$, there exists $F\in\mathcal{F}$ such that $F\cap\{x,y\}=\{x\}$. For a fixed $t, 2\le t\le 5$ and $n\rightarrow\infty$, we establish asymptotically tight estimates for the smallest integer $s=s(n,t)$ such that every family $\mathcal{F}$ with $|\mathcal{F}|\ge s$ is $t$-separable.
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.