Related papers: A Pseudoline Counterexample to the Strong Dirac Co…

We prove that pseudoholomorphic curves intersect complex 2-cycles positively in almost complex 4-manifolds. This makes possible a general and conceptually simple proof that an almost complex 4-manifold with many curves admits a taming…

In this note, we show that real line arrangements of type at most one, admitting only intersection points of multiplicity at most five, satisfy certain boundedness properties. In particular, we prove that a free real arrangement of $d$…

We study families $\mathcal{F}\subseteq 2^{[n]}$ with restricted intersections and prove a conjecture of Snevily in a stronger form for large $n$. We also obtain stability results for Kleitman's isodiametric inequality and families with…

In the present note we provide a partial classification of nearly free conic-line arrangements in the complex plane having nodes, tacnodes, and ordinary triple points. In this setting, our theoretical bound tells us that the degree of such…

We study the structure of planar point sets that determine a small number of distinct distances. Specifically, we show that if a set P of n points determines o(n) distinct distances, then no line contains \Omega(n^{7/8}) points of P and no…

Let X be a pseudocyclic association scheme in which all the nontrivial relations are strongly regular graphs with the same eigenvalues. We prove that the principal part of the first eigenmatrix of X is a linear combination of an incidence…

In this note we count linear arrangements that avoid certain patterns and show their connection to the derangement numbers. We discuss the sequence Dn, which counts linear arrangements that avoid patterns 12, 23, ..., (n-1)n, n1, and show…

Let $\mathcal{S}$ be a set of $n$ points in real four-dimensional space, no four coplanar and spanning the whole space. We prove that if the number of solids incident with exactly four points of $\mathcal{S}$ is less than $Kn^3$ for some…

We consider the system of $N$ points on the segment of the real line with the nearest-neighbor Coulomb repulsive interaction and external force $F$. For the fixed points of such systems (fixed configurations) we study the asymptotics (in…

In this paper, we consider conic-line arrangements that arise from Poncelet's closure theorem. We study unramified double covers of the union of two conics, that are induced by a $2m$-sided Poncelet transverse. As an application, we show…

We construct a model in which the splitting number is large and every ultrafilter has a small subset with no pseudo-intersection.

The famous Szemer\'{e}di-Trotter theorem states that any arrangement of $n$ points and $n$ lines in the plane determines $O(n^{4/3})$ incidences, and this bound is tight. In this paper, we prove the following Tur\'an-type result for…

In 2014, Payne-Wood proved that every non-collinear set $P$ of $n$ points in the Euclidean plane contains a point in at least $\dfrac{n}{37}$ lines determined by $P.$ This is a remarkable answer for the conjecture, which was proposed by…

A quadrisecant of a knot is a straight line intersecting the knot at four points. If a knot has finitely many quadrisecants, one can replace each subarc between two adjacent secant points by the line segment between them to get the…

We construct a sequence of subset partition graphs satisfying the dimension reduction, adjacency, strong adjacency, and endpoint count properties whose diameter has a superlinear asymptotic lower bound. These abstractions of polytope graphs…

A family ${\mathcal A} \subset {\mathcal P} [n]$ is said to be an antichain if $A \not \subset B$ for all distinct $A,B \in {\mathcal A}$. A classic result of Sperner shows that such families satisfy $|{\mathcal A}| \leq \binom {n}{\lfloor…

Using cyclotomic classes of order twelve for certain finite fields, we construct an infinite family of almost difference sets and normally regular graphs applying the theory of cyclotomy. We show that in each of these fields neither the…

We study the stability of Dirac semimetals with $N$ nodes in three spatial dimensions against strong $1/r$ Coulomb interactions. We particularly study the cases of $N=4$ and $N=16$, where the $N=4$ Dirac semimetal is described by the…

In this paper, we derive a tight upper bound for the size of an intersecting $k$-Sperner family of subspaces of the $n$-dimensional vector space $\mathbb{F}_{q}^{n}$ over finite field $\mathbb{F}_{q}$ which gives a $q$-analogue of the…

Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is…