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

We show that for any large $n$, there exists a set of $n$ points in the plane with $O(n^2/\sqrt{\log n})$ distinct distances, such that any four points in the set determine at least five distinct distances. This answers (in the negative) a…

There are many physically interesting superconformal gauge theories in four dimensions. In this talk I discuss a common phenomenon in these theories: the existence of continuous families of infrared fixed points. Well-known examples include…

The syntactic structure of sentences exhibits a striking regularity: dependencies tend to not cross when drawn above the sentence. We investigate two competing explanations. The traditional hypothesis is that this trend arises from an…

The symmetric case of the Szemer\'edi-Trotter theorem says that any configuration of $N$ lines and $N$ points in the plane has at most $O(N^{4/3})$ incidences. We describe a recipe involving just $O(N^{1/3})$ parameters which sometimes…

One of the open questions in the geometry of line arrangements is to what extent does the incidence lattice of an arrangement determine its fundamental group. Line arrangements of up to 6 lines were recently classified by K.M. Fan, and it…

We prove the "Sullivan Conjecture" on the classification of 4-dimensional complete intersections up to diffeomorphism. Here an $n$-dimensional complete intersection is a smooth complex variety formed by the transverse intersection of $k$…

We prove that a distance-regular graph with intersection array {56,36,9;1,3,48} does not exist. This intersection array is from the table of feasible parameters for distance-regular graphs in "Distance-regular graphs"\ by A.E. Brouwer, A.M.…

A generalized Davenport-Schinzel sequence is one over a finite alphabet that contains no subsequences isomorphic to a fixed forbidden subsequence. One of the fundamental problems in this area is bounding (asymptotically) the maximum length…

We consider the following problem: Given a set $S$ of $n$ distinct points in the plane, how many edge-disjoint plane straight-line spanning paths can be drawn on $S$? Each spanning path must be crossing-free, but edges from different paths…

Let $G$ be a non-abelian finite simple group. In addition, let $\Delta_G$ be the intersection graph of $G$, whose vertices are the proper nontrivial subgroups of $G$, with distinct subgroups joined by an edge if and only if they intersect…

Upper bounds to the size of a family of subsets of an n-element set that avoids certain configurations are proved. These forbidden configurations can be described by inclusion patterns and some sets having the same size. Our results are…

A family of subsets of $[n]$ is intersecting if every pair of its sets intersects. Determining the structure of large intersecting families is a central problem in extremal combinatorics. Frankl-Kupavskii and Balogh-Das-Liu-Sharifzadeh-Tran…

We give a simple argument showing that the number of edges in the intersection graph $G$ of a family of $n$ sets in the plane with a linear union-complexity is $O(\omega(G)n)$. In particular, we prove $\chi(G)\leq \text{col}(G)<…

We consider incidences among colored sets of lines in $\mathbb{R}^d$ and examine whether the existence of certain concurrences between lines of $k$ colors force the existence of at least one concurrence between lines of $k+1$ colors. This…

Let $\mathcal{Q}_1$ and $\mathcal{Q}_2$ be two arbitrary quadrics with no common hyperplane in ${\mathbb{P}}^n(\mathbb{F}_q)$. We give the best upper bound for the number of points in the intersection of these two quadrics. Our result…

We construct families of curves which provide counterexamples for a uniform boundedness question. These families generalize those studied previously by several authors. We show, in detail, what fails in the argument of Caporaso, Harris,…

Finding spanning structures with many distinct colours in properly edge-coloured graphs is a central theme in extremal combinatorics. A classical result of Andersen shows that every proper edge-colouring of the complete graph $K_n$ contains…

Classical strong external difference families (SEDFs) are much-studied combinatorial structures motivated by information security applications; it is conjectured that only one classical abelian SEDF exists with more than two sets. Recently,…

We study special linear systems of surfaces of $\mathbb{P}^3$ interpolating nine points in general position having a quadric as fixed component. By performing degenerations in the blown-up space, we interpret the quadric obstruction in…

Let $n$, $r$, $k_1,\ldots,k_r$ and $t$ be positive integers with $r\geq 2$, and $\mathcal{F}_i\ (1\leq i\leq r)$ a family of $k_i$-subsets of an $n$-set $V$. The families $\mathcal{F}_1,\ \mathcal{F}_2,\ldots,\mathcal{F}_r$ are said to be…