Ramsey numbers of ordered graphs

An ordered graph is a pair $\mathcal{G}=(G,\prec)$ where $G$ is a graph and $\prec$ is a total ordering of its vertices. The ordered Ramsey number $\overline{R}(\mathcal{G})$ is the minimum number $N$ such that every ordered complete graph with $N$ vertices and with edges colored by two colors contains a monochromatic copy of $\mathcal{G}$. In contrast with the case of unordered graphs, we show that there are arbitrarily large ordered matchings $\mathcal{M}_n$ on $n$ vertices for which $\overline{R}(\mathcal{M}_n)$ is superpolynomial in $n$. This implies that ordered Ramsey numbers of the same graph can grow superpolynomially in the size of the graph in one ordering and remain linear in another ordering. We also prove that the ordered Ramsey number $\overline{R}(\mathcal{G})$ is polynomial in the number of vertices of $\mathcal{G}$ if the bandwidth of $\mathcal{G}$ is constant or if $\mathcal{G}$ is an ordered graph of constant degeneracy and constant interval chromatic number. The first result gives a positive answer to a question of Conlon, Fox, Lee, and Sudakov. For a few special classes of ordered paths, stars or matchings, we give asymptotically tight bounds on their ordered Ramsey numbers. For so-called monotone cycles we compute their ordered Ramsey numbers exactly. This result implies exact formulas for geometric Ramsey numbers of cycles introduced by K\'arolyi, Pach, T\'oth, and Valtr.Comment: 29 pages, 13 figures, to appear in Electronic Journal of Combinatoric

Induced Ramsey-type results and binary predicates for point sets

Let $k$ and $p$ be positive integers and let $Q$ be a finite point set in general position in the plane. We say that $Q$ is $(k,p)$-Ramsey if there is a finite point set $P$ such that for every $k$-coloring $c$ of $\binom{P}{p}$ there is a subset $Q'$ of $P$ such that $Q'$ and $Q$ have the same order type and $\binom{Q'}{p}$ is monochromatic in $c$. Ne\v{s}et\v{r}il and Valtr proved that for every $k \in \mathbb{N}$, all point sets are $(k,1)$-Ramsey. They also proved that for every $k \ge 2$ and $p \ge 2$, there are point sets that are not $(k,p)$-Ramsey. As our main result, we introduce a new family of $(k,2)$-Ramsey point sets, extending a result of Ne\v{s}et\v{r}il and Valtr. We then use this new result to show that for every $k$ there is a point set $P$ such that no function $\Gamma$ that maps ordered pairs of distinct points from $P$ to a set of size $k$ can satisfy the following "local consistency" property: if $\Gamma$ attains the same values on two ordered triples of points from $P$, then these triples have the same orientation. Intuitively, this implies that there cannot be such a function that is defined locally and determines the orientation of point triples.Comment: 22 pages, 3 figures, final version, minor correction

Monotone crossing number of complete graphs

In 1958, Hill conjectured that the minimum number of crossings in a drawing of Kn is exactly Z(n) = 1/4 n-1/2/2 n−2/2 n−3/2. Generalizing the result by Ábrego et al. for 2-page book drawings, we prove this conjecture for plane drawings in which edges are represented by x-monotone curves. In fact, our proof shows that the conjecture remains true for xmonotone drawings in which adjacent edges do not cross and we count only pairs of edges which cross odd number of times. We also discuss a combinatorial characterization of these drawings.European Science Foundatio

Management and imaging of bronchopulmonary malformations in children

Background Bronchopulmonary malformations (BPMs) consists of a broad spectrum of developmental abnormalities, ranging from abnormal lung with normal vasculature to abnormal vasculature with normal lungs and lesions with both parenchymal and vascular abnormalities. Terminology remains a problem, the use of descriptive approach is strongly supported in practice. The aim of our study was to assess the accuracy of computed tomography (CT) and magnetic resonance imaging (MRI) in classifying the different types of BPMs and to correlate this imaging with pathologic finding. Materials and methods We identified 24 patients reffered to our institution between years 2010 and 2015 with prenatal ultrasound (US) diagnosis of BPM, who had undergone surgical resection. Postnatal chest CT scans and fetal MRI of these patients formed the basis of our retrospective study. Two radiologists blinded to the histopathological results rewieved all CT and fetal MRI scans. Detected lesions were classified by predetermined criteria. These data were correlated with histopathological results. Results The significant diagnosis overlap of 71% (17 cases out of 24) between the CT and histology was observed. The significant diagnosis overlap between MRI and histology was observed of 80 % (8 cases out of 10). The diagnosis overlap of..

Bounds for Pach's Selection Theorem and for the Minimum Solid Angle in a Simplex

We estimate the selection constant in the following geometric selection theorem by Pach: For every positive integer d, there is a constant $$c_d > 0$$ c d > 0 such that whenever $$X_1, \ldots , X_{d+1}$$ X 1 , ... , X d + 1 are n-element subsets of $$\mathbb {R}^d$$ R d , we can find a point $${\mathbf {p}}\in \mathbb {R}^d$$ p ∈ R d and subsets $$Y_i \subseteq X_i$$ Y i ⊆ X i for every $$i \in [d+1]$$ i ∈ [ d + 1 ] , each of size at least $$c_d n$$ c d n , such that $${\mathbf {p}}$$ p belongs to all rainbow d-simplices determined by $$Y_1, \ldots , Y_{d+1}$$ Y 1 , ... , Y d + 1 , i.e., simplices with one vertex in each $$Y_i$$ Y i . We show a super-exponentially decreasing upper bound $$c_d\le e^{-(1/2-o(1))(d \ln d)}$$ c d ≤ e - ( 1 / 2 - o ( 1 ) ) ( d ln d ) . The ideas used in the proof of the upper bound also help us to prove Pach's theorem with $$c_d \ge 2^{-2^{d^2 + O(d)}}$$ c d ≥ 2 - 2 d 2 + O ( d ) , which is a lower bound doubly exponentially decreasing in d (up to some polynomial in the exponent). For comparison, Pach's original approach yields a triply exponentially decreasing lower bound. On the other hand, Fox, Pach, and Suk recently obtained a hypergraph density result implying a proof of Pach's theorem with $$c_d \ge 2^{-O(d^2\log d)}$$ c d ≥ 2 - O ( d 2 log d ) . In our construction for the upper bound, we use the fact that the minimum solid angle of every d-simplex is super-exponentially small. This fact was previously unknown and might be of independent interest. For the lower bound, we improve the ‘separation' part of the argument by showing that in one of the key steps only $$d+1$$ d + 1 separations are necessary, compared to $$2^d$$ 2 d separations in the original proof. We also provide a measure version of Pach's theorem