Two-part and k-Sperner families: New proofs using permutations

This is a paper about the beauty of the permutation method. New and shorter proofs are given for the theorem [P. L. Erdős and G. O. H. Katona, J. Combin. Theory. Ser. A,4

The history of degenerate (bipartite) extremal graph problems

This paper is a survey on Extremal Graph Theory, primarily focusing on the case when one of the excluded graphs is bipartite. On one hand we give an introduction to this field and also describe many important results, methods, problems, and constructions.Comment: 97 pages, 11 figures, many problems. This is the preliminary version of our survey presented in Erdos 100. In this version 2 only a citation was complete

Covering Partial Cubes with Zones

A partial cube is a graph having an isometric embedding in a hypercube. Partial cubes are characterized by a natural equivalence relation on the edges, whose classes are called zones. The number of zones determines the minimal dimension of a hypercube in which the graph can be embedded. We consider the problem of covering the vertices of a partial cube with the minimum number of zones. The problem admits several special cases, among which are the problem of covering the cells of a line arrangement with a minimum number of lines, and the problem of finding a minimum-size fibre in a bipartite poset. For several such special cases, we give upper and lower bounds on the minimum size of a covering by zones. We also consider the computational complexity of those problems, and establish some hardness results

Induction of Crystallization of Calcium Oxalate Dihydrate in Micellar Solutions of Anionic Surfactants

Calcium oxalate dihydrate (CaC2O4.(2+x)H2O; COD; x ≤ 0.5) does not readily crystallize from electrolytic solutions but appears as a component in crystalluria. In this paper, we review in vitro studies on the factors responsible for its nucleation and growth with special attention given to the role of surfactants. The following surfactants were tested: dodecyl ammonium chloride (cationic), octaethylene monohexadecylether (non-ionic), sodium dodecyl sulfate (SOS, anionic), dioctyl sulphosuccinate (AOT, anionic), and sodium cholate (NaC, anionic). The cationic and some of the anionic surfactants (SOS, AOT) induced different habit modifications of growing calcium oxalate crystals by preferential adsorption at different crystal faces. In addition, the anionic surfactants effectively induced crystallization of COD at the expense of COM, the proportion of COD in the precipitates abruptly increasing above a critical surfactant concentration, close to, but not necessarily identical with the respective CMC. A mechanism is proposed, whereby crystallization of COD in the presence of surfactants is a consequence of the inhibition of COM by preferential adsorption of surfactant hemimicelles (two-dimensional surface aggregates) at the surfaces of growing crystals

On Arrangements of Orthogonal Circles

In this paper, we study arrangements of orthogonal circles, that is, arrangements of circles where every pair of circles must either be disjoint or intersect at a right angle. Using geometric arguments, we show that such arrangements have only a linear number of faces. This implies that orthogonal circle intersection graphs have only a linear number of edges. When we restrict ourselves to orthogonal unit circles, the resulting class of intersection graphs is a subclass of penny graphs (that is, contact graphs of unit circles). We show that, similarly to penny graphs, it is NP-hard to recognize orthogonal unit circle intersection graphs.Comment: Appears in the Proceedings of the 27th International Symposium on Graph Drawing and Network Visualization (GD 2019

Subtended angles

The first is partially supported by NSF grant DMS 1301614. The second author is partially supported by NSF grant DMS 1301614 and MULTIPLEX no. 317532. The third author’s research supported in part by the Hungarian National Science Foundation OTKA 104343, by the Simons Foundation Collaboration Grant #317487, and by the European Research Council Advanced Investigators Grant 267195

Every Large Point Set contains Many Collinear Points or an Empty Pentagon

We prove the following generalised empty pentagon theorem: for every integer $\ell \geq 2$, every sufficiently large set of points in the plane contains $\ell$ collinear points or an empty pentagon. As an application, we settle the next open case of the "big line or big clique" conjecture of K\'ara, P\'or, and Wood [\emph{Discrete Comput. Geom.} 34(3):497--506, 2005]

The lacuna of capital, the state and war? The lost global history and theory of Eastern agency

In this article I seek to constructively engage Alex Anievas’s seminal book that is deservedly the subject of this forum. For Anievas has become a key figure in the revival of Trotskyism in IR and his is one of the first book-length treatments of the New Trotskyist theory of the international. My critique is meant merely as a constructive effort to push his excellent scholarship further in terms of developing his non-Eurocentric approach. In the first section I argue that his book represents a giant leap forward for the New Trotskyist IR. However, in the following sections I argue that although undeniably a brave attempt nevertheless, in the last instance, Anievas falls a few steps short in realising a genuinely non-Eurocentric account of world politics. This is because while he certainly restores or brings in ‘the lost theory and history of IR’ that elevates class forces to a central role in shaping world politics, nevertheless he fails to bring in ‘the lost global theory and history of Eastern agency’ that constitutes, in my view, the key ingredient of a non-Eurocentric approach to world politics. I also argue that while his anti-reductionist ontological credentials are for the most part extremely impressive, nevertheless, I argue that these are compromised in his analysis of Hitler’s racism. Finally, in the conclusion I ask whether the theoretical architecture of the New Trotskyism in IR is capable of developing a non-Eurocentric approach before concluding in the affirmative with respect to its modern revisionist incarnation of which Anievas is in the vanguard

On Eigenvalues of Random Complexes

We consider higher-dimensional generalizations of the normalized Laplacian and the adjacency matrix of graphs and study their eigenvalues for the Linial-Meshulam model $X^k(n,p)$ of random $k$-dimensional simplicial complexes on $n$ vertices. We show that for $p=\Omega(\log n/n)$, the eigenvalues of these matrices are a.a.s. concentrated around two values. The main tool, which goes back to the work of Garland, are arguments that relate the eigenvalues of these matrices to those of graphs that arise as links of $(k-2)$-dimensional faces. Garland's result concerns the Laplacian; we develop an analogous result for the adjacency matrix. The same arguments apply to other models of random complexes which allow for dependencies between the choices of $k$-dimensional simplices. In the second part of the paper, we apply this to the question of possible higher-dimensional analogues of the discrete Cheeger inequality, which in the classical case of graphs relates the eigenvalues of a graph and its edge expansion. It is very natural to ask whether this generalizes to higher dimensions and, in particular, whether the higher-dimensional Laplacian spectra capture the notion of coboundary expansion - a generalization of edge expansion that arose in recent work of Linial and Meshulam and of Gromov. We show that this most straightforward version of a higher-dimensional discrete Cheeger inequality fails, in quite a strong way: For every $k\geq 2$ and $n\in \mathbb{N}$, there is a $k$-dimensional complex $Y^k_n$ on $n$ vertices that has strong spectral expansion properties (all nontrivial eigenvalues of the normalised $k$-dimensional Laplacian lie in the interval $[1-O(1/\sqrt{n}),1+O(1/\sqrt{n})]$) but whose coboundary expansion is bounded from above by $O(\log n/n)$ and so tends to zero as $n\rightarrow \infty$; moreover, $Y^k_n$ can be taken to have vanishing integer homology in dimension less than $k$.Comment: Extended full version of an extended abstract that appeared at SoCG 2012, to appear in Israel Journal of Mathematic