Overlap properties of geometric expanders

The {\em overlap number} of a finite $(d+1)$-uniform hypergraph $H$ is defined as the largest constant $c(H)\in (0,1]$ such that no matter how we map the vertices of $H$ into $\R^d$, there is a point covered by at least a $c(H)$-fraction of the simplices induced by the images of its hyperedges. In~\cite{Gro2}, motivated by the search for an analogue of the notion of graph expansion for higher dimensional simplicial complexes, it was asked whether or not there exists a sequence $\{H_n\}_{n=1}^\infty$ of arbitrarily large $(d+1)$-uniform hypergraphs with bounded degree, for which $\inf_{n\ge 1} c(H_n)>0$. Using both random methods and explicit constructions, we answer this question positively by constructing infinite families of $(d+1)$-uniform hypergraphs with bounded degree such that their overlap numbers are bounded from below by a positive constant $c=c(d)$. We also show that, for every $d$, the best value of the constant $c=c(d)$ that can be achieved by such a construction is asymptotically equal to the limit of the overlap numbers of the complete $(d+1)$-uniform hypergraphs with $n$ vertices, as $n\rightarrow\infty$. For the proof of the latter statement, we establish the following geometric partitioning result of independent interest. For any $d$ and any $\epsilon>0$, there exists $K=K(\epsilon,d)\ge d+1$ satisfying the following condition. For any $k\ge K$, for any point $q \in \mathbb{R}^d$ and for any finite Borel measure $\mu$ on $\mathbb{R}^d$ with respect to which every hyperplane has measure $0$, there is a partition $\mathbb{R}^d=A_1 \cup \ldots \cup A_{k}$ into $k$ measurable parts of equal measure such that all but at most an $\epsilon$-fraction of the $(d+1)$-tuples $A_{i_1},\ldots,A_{i_{d+1}}$ have the property that either all simplices with one vertex in each $A_{i_j}$ contain $q$ or none of these simplices contain $q$

Q2Q_2-free families in the Boolean lattice

For a family $\mathcal{F}$ of subsets of [n]=\{1, 2, ..., n} ordered by inclusion, and a partially ordered set P, we say that $\mathcal{F}$ is P-free if it does not contain a subposet isomorphic to P. Let $ex(n, P)$ be the largest size of a P-free family of subsets of [n]. Let $Q_2$ be the poset with distinct elements a, b, c, d, a<b, c<d; i.e., the 2-dimensional Boolean lattice. We show that $2N -o(N) \leq ex(n, Q_2)\leq 2.283261N +o(N),$ where $N = \binom{n}{\lfloor n/2 \rfloor}$. We also prove that the largest $Q_2$-free family of subsets of [n] having at most three different sizes has at most 2.20711N members.Comment: 18 pages, 2 figure

Superpatterns and Universal Point Sets

An old open problem in graph drawing asks for the size of a universal point set, a set of points that can be used as vertices for straight-line drawings of all n-vertex planar graphs. We connect this problem to the theory of permutation patterns, where another open problem concerns the size of superpatterns, permutations that contain all patterns of a given size. We generalize superpatterns to classes of permutations determined by forbidden patterns, and we construct superpatterns of size n^2/4 + Theta(n) for the 213-avoiding permutations, half the size of known superpatterns for unconstrained permutations. We use our superpatterns to construct universal point sets of size n^2/4 - Theta(n), smaller than the previous bound by a 9/16 factor. We prove that every proper subclass of the 213-avoiding permutations has superpatterns of size O(n log^O(1) n), which we use to prove that the planar graphs of bounded pathwidth have near-linear universal point sets.Comment: GD 2013 special issue of JGA

Tur\'an numbers for Ks,tK_{s,t}-free graphs: topological obstructions and algebraic constructions

We show that every hypersurface in $\R^s\times \R^s$ contains a large grid, i.e., the set of the form $S\times T$, with $S,T\subset \R^s$. We use this to deduce that the known constructions of extremal $K_{2,2}$-free and $K_{3,3}$-free graphs cannot be generalized to a similar construction of $K_{s,s}$-free graphs for any $s\geq 4$. We also give new constructions of extremal $K_{s,t}$-free graphs for large $t$.Comment: Fixed a small mistake in the application of Proposition

Proposed Revision to the Taxonomy of the Genus Pestivirus; Family Flaviviridae

We propose the creation of seven new species in the genus Pestivirus (family Flaviviridae) in addition to the four existing species, and naming species in a host-independent manner using the format Pestivirus X. Only the virus species names would change; virus isolates would still be referred to by their original names. The original species would be re-designated as Pestivirus A (original designation Bovine viral diarrhea virus 1), Pestivirus B (Bovine viral diarrhea virus 2), Pestivirus C (Classical swine fever virus) and Pestivirus D (Border disease virus). The seven new species (and example isolates) would be Pestivirus E (pronghorn pestivirus), Pestivirus F (Bungowannah virus), Pestivirus G (giraffe pestivirus), Pestivirus H (Hobi-like pestivirus), Pestivirus I (Aydin-like pestivirus), Pestivirus J (rat pestivirus) and Pestivirus K (atypical porcine pestivirus). A bat-derived virus and pestiviruses identified from sheep and goat (Tunisian sheep pestiviruses), which lack complete coding region sequences, may represent two additional species

Universal Geometric Graphs

We introduce and study the problem of constructing geometric graphs that have few vertices and edges and that are universal for planar graphs or for some sub-class of planar graphs; a geometric graph is \emph{universal} for a class $\mathcal H$ of planar graphs if it contains an embedding, i.e., a crossing-free drawing, of every graph in $\mathcal H$. Our main result is that there exists a geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex forests; this extends to the geometric setting a well-known graph-theoretic result by Chung and Graham, which states that there exists an $n$-vertex graph with $O(n \log n)$ edges that contains every $n$-vertex forest as a subgraph. Our $O(n \log n)$ bound on the number of edges cannot be improved, even if more than $n$ vertices are allowed. We also prove that, for every positive integer $h$, every $n$-vertex convex geometric graph that is universal for $n$-vertex outerplanar graphs has a near-quadratic number of edges, namely $\Omega_h(n^{2-1/h})$; this almost matches the trivial $O(n^2)$ upper bound given by the $n$-vertex complete convex geometric graph. Finally, we prove that there exists an $n$-vertex convex geometric graph with $n$ vertices and $O(n \log n)$ edges that is universal for $n$-vertex caterpillars.Comment: 20 pages, 8 figures; a 12-page extended abstracts of this paper will appear in the Proceedings of the 46th Workshop on Graph-Theoretic Concepts in Computer Science (WG 2020

Expanded classification of hepatitis C Virus into 7 genotypes and 67 Subtypes:updated criteria and assignment web resource

The 2005 consensus proposal for the classification of hepatitis C virus (HCV) presented an agreed and uniform nomenclature for HCV variants and the criteria for their assignment into genotypes and subtypes. Since its publication, the available dataset of HCV sequences has vastly expanded through advancement in nucleotide sequencing technologies and an increasing focus on the role of HCV genetic variation in disease and treatment outcomes. The current study represents a major update to the previous consensus HCV classification, incorporating additional sequence information derived from over 1,300 (near-)complete genome sequences of HCV available on public databases in May 2013. Analysis resolved several nomenclature conflicts between genotype designations and using consensus criteria created a classification of HCV into seven confirmed genotypes and 67 subtypes. There are 21 additional complete coding region sequences of unassigned subtype. The study additionally describes the development of a Web resource hosted by the International Committee for Taxonomy of Viruses (ICTV) that maintains and regularly updates tables of reference isolates, accession numbers, and annotated alignments (http://talk.ictvonline.org/links/hcv/hcv-classification.htm). The Flaviviridae Study Group urges those who need to check or propose new genotypes or subtypes of HCV to contact the Study Group in advance of publication to avoid nomenclature conflicts appearing in the literature. While the criteria for assigning genotypes and subtypes remain unchanged from previous consensus proposals, changes are proposed in the assignment of provisional subtypes, subtype numbering beyond “w,” and the nomenclature of intergenotypic recombinant. Conclusion: This study represents an important reference point for the consensus classification of HCV variants that will be of value to researchers working in clinical and basic science fields. (Hepatology 2014;59:318-327

Glycan shifting on hepatitis C virus (HCV) E2 glycoprotein is a mechanism for escape from broadly neutralizing antibodies

Hepatitis C virus (HCV) infection is a major cause of liver disease and hepatocellular carcinoma. Glycan shielding has been proposed to be a mechanism by which HCV masks broadly neutralizing epitopes on its viral glycoproteins. However, the role of altered glycosylation in HCV resistance to broadly neutralizing antibodies is not fully understood. Here, we have generated potent HCV neutralizing antibodies hu5B3.v3 and MRCT10.v362 that, similar to the previously described AP33 and HCV1, bind to a highly conserved linear epitope on E2. We utilize a combination of in vitro resistance selections using the cell culture infectious HCV and structural analyses to identify mechanisms of HCV resistance to hu5B3.v3 and MRCT10.v362. Ultra deep sequencing from in vitro HCV resistance selection studies identified resistance mutations at asparagine N417 (N417S, N417T and N417G) as early as 5 days post treatment. Comparison of the glycosylation status of soluble versions of the E2 glycoprotein containing the respective resistance mutations revealed a glycosylation shift from N417 to N415 in the N417S and N417T E2 proteins. The N417G E2 variant was glycosylated neither at residue 415 nor at residue 417 and remained sensitive to MRCT10.v362. Structural analyses of the E2 epitope bound to hu5B3.v3 Fab and MRCT10.v362 Fab using X-ray crystallography confirmed that residue N415 is buried within the antibody–peptide interface. Thus, in addition to previously described mutations at N415 that abrogate the β-hairpin structure of this E2 linear epitope, we identify a second escape mechanism, termed glycan shifting, that decreases the efficacy of broadly neutralizing HCV antibodies