Mathematics

Given finite configurations P 1 ,?? P n ??R d , let us denote by m R d ( P 1 ,?? P n ) the maximum density a set A??R d can have without containing congruent copies of any P i . In this paper we will study this geometrical parameter, called the independence density of the considered configurations, and give several results we believe are interesting. For instance we show that, under suitable size and non-degeneracy conditions, m R d ( t 1 P 1 , t 2 P 2 ,?? t n P n ) tends to ??n i=1 m R d ( P i ) as the ratios between consecutive dilation parameters t i+1 / t i grow large; this shows an exponential decay on the density when forbidding multiple dilates of a given configuration, and gives a common generalization of theorems by Bourgain and by Bukh in geometric Ramsey theory. We also consider the analogous parameter m S d ( P 1 ,?? P n ) on the more complicated framework of sets on the unit sphere S d , obtaining the corresponding results in this setting.

The unit distance graph G 1 R d is the infinite graph whose nodes are points in R d , with an edge between two points if the Euclidean distance between these points is 1. The 2-dimensional version G 1 R 2 of this graph is typically studied for its chromatic number, as in the Hadwiger-Nelson problem. However, other properties of unit distance graphs are rarely studied. Here, we consider the restriction of G 1 R d to closed convex subsets X of R d . We show that the graph G 1 R d [X] is connected precisely when the radius of r(X) of X is equal to 0, or when r(X)?? and the affine dimension of X is at least 2. For hyperrectangles, we give bounds for the graph diameter in the critical case that the radius is exactly 1.

We investigate (0,1) -matrices that are {\em convex}, which means that the ones are consecutive in every row and column. These matrices occur in discrete tomography. The notion of ranked essential sets, known for permutation matrices, is extended to convex sets. We show a number of results for the class $\mc{C}(R,S)$ of convex matrices with given row and column sum vectors R and S . Also, it is shown that the ranked essential set uniquely determines a matrix in $\mc{C}(R,S)$.

The directions of an infinite graph G are a tangle-like description of its ends: they are choice functions that choose compatibly for all finite vertex sets X?�V(G) a component of G?�X . Although every direction is induced by a ray, there exist directions of graphs that are not uniquely determined by any countable subset of their choices. We characterise these directions and their countably determined counterparts in terms of star-like substructures or rays of the graph. Curiously, there exist graphs whose directions are all countably determined but which cannot be distinguished all at once by countably many choices. We structurally characterise the graphs whose directions can be distinguished all at once by countably many choices, and we structurally characterise the graphs which admit no such countably many choices. Our characterisations are phrased in terms of normal trees and tree-decompositions. Our four (sub)structural characterisations imply combinatorial characterisations of the four classes of infinite graphs that are defined by the first and second axiom of countability applied to their end spaces: the two classes of graphs whose end spaces are first countable or second countable, respectively, and the complements of these two classes.

In 1930, Kuratowski showed that K 3,3 and K 5 are the only two minor-minimal non-planar graphs. Robertson and Seymour extended finiteness of the set of forbidden minors for any surface. ?irá? and Kochol showed that there are infinitely many k -crossing-critical graphs for any k?? , even if restricted to simple 3 -connected graphs. Recently, 2 -crossing-critical graphs have been completely characterized by Bokal, Oporowski, Richter, and Salazar. We present a simplified description of large 2-crossing-critical graphs and use this simplification to count Hamiltonian cycles in such graphs. We generalize this approach to an algorithm counting Hamiltonian cycles in all 2-tiled graphs, thus extending the results of Bodroža-Panti?, Kwong, Doroslova?ki, and Panti? for n=2 .

A Schröder path is a lattice path from (0,0) to (2n,0) with steps (1,1) , (1,−1) and (2,0) that never goes below the x− axis. A small Schröder path is a Schröder path with no (2,0) steps on the x− axis. In this paper, a 3-variable generating function R L (x,y,z) is given for Schröder paths and small Schröder paths respectively. As corollaries, we obtain the generating functions for several kinds of generalized Schröder paths counted according to the order in a unified way.

Let S k (n) be the maximum number of orientations of an n -vertex graph G in which no copy of K k is strongly connected. For all integers n , k?? where n?? or k?? , we prove that S k (n)= 2 t k?? (n) , where t k?? (n) is the number of edges of the n -vertex (k??) -partite Turán graph T k?? (n) , and that T k?? (n) is the only n -vertex graph with this number of orientations. Furthermore, S 4 (4)=40 and this maximality is achieved only by K 4 .

For a planar graph H , let N P (n,H) denote the maximum number of copies of H in an n -vertex planar graph. In this paper, we prove that N P (n, P 7 )??4 27 n 4 , N P (n, C 6 )??n/3 ) 3 , N P (n, C 8 )??n/4 ) 4 and N P (n, K 4 {1})??n/6 ) 6 , where K 4 {1} is the 1 -subdivision of K 4 . In addition, we obtain significantly improved upper bounds on N P (n, P 2m+1 ) and N P (n, C 2m ) for m?? . For a wide class of graphs H , the key technique developed in this paper allows us to bound N P (n,H) in terms of an optimization problem over weighted graphs.

In this paper we consider the hyperplane arrangement in R n whose hyperplanes are { x i + x j =1???�i<j?�n}?�{ x i =0,1???�i?�n} . We call it the \emph{boxed threshold arrangement} since we show that the bounded regions of this arrangement are contained in an n -cube and are in one-to-one correspondence with the labeled threshold graphs on n vertices. The problem of counting regions of this arrangement was studied earlier by Joungmin Song. He determined the characteristic polynomial of this arrangement by relating its coefficients to the count of certain graphs. Here, we provide bijective arguments to determine the number of regions. In particular, we construct certain signed partitions of the set {?�n,??n}?�{0} and also construct colored threshold graphs on n vertices and show that both these objects are in bijection with the regions of the boxed threshold arrangement. We independently count these objects and provide closed form formula for the number of regions.

A digraph is {\em d -dominating} if every set of at most d vertices has a common out-neighbor. For all integers d?? , let f(d) be the smallest integer such that the vertices of every 2-edge-colored (finite or infinite) complete digraph (including loops) can be covered by the vertices of at most f(d) monochromatic d -dominating subgraphs. Note that the existence of f(d) is not obvious -- indeed, the question which motivated this paper was simply to determine whether f(d) is bounded, even for d=2 . We answer this question affirmatively for all d?? , proving 4?�f(2)?? and 2d?�f(d)??d( d d ?? d?? ) for all d?? . We also give an example to show that there is no analogous bound for more than two colors. Our result provides a positive answer to a question regarding an infinite analogue of the Burr-Erd?s conjecture on the Ramsey numbers of d -degenerate graphs. Moreover, a special case of our result is related to properties of d -paradoxical tournaments.