기초과학연구원 이산수학그룹
Suppose that $E$ is a subset of $\mathbb{F}_q^n$, so that each point is contained in $E$ with probability $\theta$, independently of all other points. Then, what is the probability that there is an $m$-dimensional affine subspace that contains at least $\ell$ points of $E$? What is the probability that $E$ intersects all $m$-dimensional affine subspaces? …
In 1975, Szemerédi proved that for every real number $\delta > 0 $ and every positive integer $k$, there exists a positive integer $N$ such that every subset $A$ of the set $\{1, 2, \cdots, N \}$ with $|A| \geq \delta N$ contains an arithmetic progression of length $k$. There has been a plethora of …
We introduce an odd coloring of a graph, which was introduced very recently, motivated by parity type colorings of graphs. A proper vertex coloring of graph $G$ is said to be odd if for each non-isolated vertex $x \in V (G)$ there exists a color $c$ such that $c$ is used an odd number of …
In this talk, we introduce homomorphisms between binary matroids that generalize graph homomorphisms. For a binary matroid $N$, we prove a complexity dichotomy for the problem $\rm{Hom}_\mathbb{M}(N)$ of deciding if a binary matroid $M$ admits a homomorphism to $N$. The problem is polynomial-time solvable if $N$ has a loop or has no circuits of odd …
The upper tail problem for subgraph counts in the Erdos-Renyi graph, introduced by Janson-Ruciński, has attracted a lot of attention. There is a class of Gibbs measures associated with subgraph counts, called exponential random graph model (ERGM). Despite its importance, lots of fundamental questions have remained unanswered owing to the lack of exact solvability. In …
Hadwiger's transversal theorem gives necessary and sufficient conditions for the existence of a line transversal to a family of pairwise disjoint convex sets in the plane. These conditions were subsequently generalized to hyperplane transversals in $\mathbb{R}^d$ by Goodman, Pollack, and Wenger. Here we establish a colorful extension of their theorem, which proves a conjecture of …
Reconfiguration is about changing instances in small steps. For example, one can perform certain moves on a Rubik's cube, each of them changing its configuration a bit. In this case, in at most 20 steps, one can end up with the preferred result. One could construct a graph with as nodes the possible configurations of …
SATNet is a differentiable constraint solver with a custom backpropagation algorithm, which can be used as a layer in a deep-learning system. It is a promising proposal for bridging deep learning and logical reasoning. In fact, SATNet has been successfully applied to learn, among others, the rules of a complex logical puzzle, such as Sudoku, …
Twin-width is a recently introduced graph parameter based on vertex contraction sequences. On classes of bounded twin-width, problems expressible in FO logic can be solved in FPT time when provided with a sequence witnessing the bound. Classes of bounded twin-width are very diverse, notably including bounded rank-width, $\Omega ( \log (n) )$-subdivisions of graphs of …
Given a set $E$ and a point $y$ in a vector space over a finite field, the radial projection $\pi_y(E)$ of $E$ from $y$ is the set of lines that through $y$ and points of $E$. Clearly, $|\pi_y(E)|$ is at most the minimum of the number of lines through $y$ and $|E|$. I will discuss …