Related papers: Some exact values of the inducibility and statisti…

Let $H$ be a graph on $h$ vertices. The number of induced copies of $H$ in a graph $G$ is denoted by $i_H(G)$. Let $i_H(n)$ denote the maximum of $i_H(G)$ taken over all graphs $G$ with $n$ vertices. Let $f(n,h) = \Pi_{i}^h a_i$ where…

A basic problem in constant dimension subspace coding is to determine the maximal possible size ${\bf A}_q(n,d,k)$ of a set of $k$-dimensional subspaces in ${\bf F}_q^n$ such that the subspace distance satisfies…

The isoperimetric problem with a density or weighting seeks to enclose prescribed weighted area with minimum weighted perimeter. According to Chambers' recent proof of the Log Convex Density Conjecture, for many densities on $\mathbb{R}^n$…

Using flag algebras, we prove that the minimum density of $8$-cliques in a large graph without an independent set of size $3$ is $491411/268435456+o(1)$, thus resolving a new case of an old problem of Erd\H{o}s [Magyar Tud. Akad. Mat.…

For any positive integer $r$, an $r$-identifying code on a graph $G$ is a set $C\subset V(G)$ such that for every vertex in $V(G)$, the intersection of the radius-$r$ closed neighborhood with $C$ is nonempty and pairwise distinct. For a…

Given a hereditary family $\mathcal{G}$ of admissible graphs and a function $\lambda(G)$ that linearly depends on the statistics of order-$\kappa$ subgraphs in a graph $G$, we consider the extremal problem of determining…

We consider the imbedding inequality || f ||_{L^r(R^d)} <= S_{r,n,d} || f ||_{H^{n}(R^d)}; H^{n}(R^d) is the Sobolev space (or Bessel potential space) of L^2 type and (integer or fractional) order n. We write down upper bounds for the…

For a graph $G$ and $e\in [0,1]$, denote by $I_G(e)$ the supremum of densities of $G$ over $n$-vertex graphs with edge density $e$ as $n$ goes to infinity. Liu, Mubayi and Reiher asked if there exists a graph $G$, where $I_G(e)$ has a…

In the area of extremal graph theory, there exists a problem that investigates the maximum induced density of a $k$-vertex graph $H$ in any $n$-vertex graph $G$. This is known as the problem of \emph{inducibility} that was first introduced…

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…

For a fixed natural number $t\geq 2$, we consider $t$-uniform random hypergraphs $\mathscr{H} (n,t,p)$ on $n$ vertices $[n]=\{1,\ldots, n\}$, where each $t$-subset of $[n]$ is included as a hyperedge with probability $p$ and independently.…

Verifying uniform conditions over continuous spaces through random sampling is fundamental in machine learning and control theory, yet classical coverage analyses often yield conservative bounds, particularly at small failure probabilities.…

We consider a variant of the densest subgraph problem in networks with single or multiple edge attributes. For example, in a social network, the edge attributes may describe the type of relationship between users, such as friends, family,…

We study the parameterized complexity of the problems of finding a maximum common (induced) subgraph of two given graphs. Since these problems generalize several NP-complete problems, they are intractable even when parameterized by strongly…

We give another version of Huang's proof that an induced subgraph of the n-dimensional cube graph containing over half the vertices has maximal degree at least $\sqrt{n}$, which implies the Sensitivity Conjecture. This argument uses…

Consider the hyperplanes at a fixed distance $t$ from the center of the hypercube $[0,1]^d$. Significant attention has been given to determining the hyperplanes $H$ among these such that the $(d-1)$-dimensional volume of $H\cap[0,1]^d$ is…

Let k>=3 be an integer, we prove that the maximum induced density of the k-vertex directed star in a directed graph is attained by an iterated blow-up construction. This confirms a conjecture by Falgas-Ravry and Vaughan, who proved this for…

We determine the asymptotic behavior of the maximum subgraph density of large random graphs with a prescribed degree sequence. The result applies in particular to the Erd\H{o}s-R\'{e}nyi model, where it settles a conjecture of Hajek [IEEE…

We use Razborov's flag algebra method to show an asymptotic upper bound for the maximal induced density $i(\vec P_3)$ of the orgraph $\vec P_3$ in an arbitrary orgraph. A conjecture of Thomass\'e states that $i(\vec P_3)=2/5$. The hitherto…

We prove that a subset of the hypercube $(0,1)^d$ with volume sufficiently close to $\frac12$ has (relative) perimeter greater than or equal to $1$. This settles a conjecture by Brezis and Bruckstein. We also prove that, in contrast with…