Related papers: The Erd\H{o}s-Ko-Rado theorem for twisted Grassman…

Let $G = (V, E)$ be a connected graph, and let $T$ in $V$ be a subset of vertices. An orientation of $G$ is called $T$-odd if any vertex $v \in V$ has odd in-degree if and only if it is in $T$. Finding a T -odd orientation of G can be…

We show that the edges of any $d$-regular graph can be almost decomposed into paths of length roughly $d$, giving an approximate solution to a problem of Kotzig from 1957. Along the way, we show that almost all of the vertices of a…

The Erd\H{o}s-Hajnal conjecture is one of the most classical and well-known problems in extremal and structural combinatorics dating back to 1977. It asserts that in stark contrast to the case of a general $n$-vertex graph if one imposes…

Celebrated results often unfold like episodes in a long-running series. In the field of extremal set thoery, Erd\H{o}s, Ko, and Rado in 1961 established that any $k$-uniform intersecting family on $[n]$ has a maximum size of…

For graphs $T, H$, let $ex(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex $H$-free graph. In this paper we prove some sharp results on this generalization of Tur\'an numbers, where our focus is for the graphs $T,H$…

The new graph parameter twin-width, introduced by Bonnet, Kim, Thomass e and Watrigant in 2020, allows for an FPT algorithm for testing all FO properties of graphs. This makes classes of efficiently bounded twin-width attractive from the…

We develop a quantitative large deviations theory for random hypergraphs, which rests on tensor decomposition and counting lemmas under a novel family of cut-type norms. As our main application, we obtain sharp asymptotics for joint upper…

While investigating odd-cycle free hypergraphs, Gy\H{o}ri and Lemons introduced a colored version of the classical theorem of Erd\H{o}s and Gallai on $P_k$-free graphs. They proved that any graph $G$ with a proper vertex coloring and no…

In 1981, Erd\H{o}s and Simonovits conjectured that for any bipartite graph $H$ we have $\mathrm{ex}(n,H)=O(n^{3/2})$ if and only if $H$ is $2$-degenerate. Later, Erd\H{o}s offered 250 dollars for a proof and 500 dollars for a…

For any finite group $G$, the equivariant Gromov-Witten invariants of $[\mathbb{C}^r/G]$ can be viewed as a certain twisted Gromov-Witten invariants of the classifying stack $\mathcal{B} G$. In this paper, we use Tseng's orbifold quantum…

In this paper, we investigate the connectivity of friends-and-strangers graphs, which were introduced by Defant and Kravitz in 2020. We begin by considering friends-and-strangers graphs arising from two random graphs and consider the…

Erd\H{o}s-Stone Theorem is a well-known result in extremal graph theory which determines the asymptotic behaviour of maximum number of edges in an $n$-vertex $H$-free graph. In 2009, Nikiforov gave a spectral version of Erd\H{o}s-Stone…

We consider two possible extensions of a theorem of Thomassen characterizing the graphs admitting a 2-vertex-connected orientation. First, we show that the problem of deciding whether a mixed graph has a 2-vertex-connected orientation is…

As a generalization of the Edmonds arborescence packing theorem, Kamiyama--Katoh--Takizawa (2009) gave a good characterization of directed graphs that contain arc-disjoint arborescences spanning the set of vertices reachable from each root.…

The concept of pseudo-distance-regularity around a vertex of a graph is a natural generalization, for non-regular graphs, of the standard distance-regularity around a vertex. In this note, we prove that a pseudo-distance-regular graph…

Komlos [Komlos: Tiling Turan Theorems, Combinatorica, 2000] determined the asymptotically optimal minimum-degree condition for covering a given proportion of vertices of a host graph by vertex-disjoint copies of a fixed graph H, thus…

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich…

In 1962, Erd\H{o}s proved a theorem on the existence of Hamilton cycles in graphs with given minimum degree and number of edges. Significantly strengthening in case of balanced bipartite graphs, Moon and Moser proved a corresponding theorem…

In 1974, Erd\H{o}s posed the following problem. Given an oriented graph $H$, determine or estimate the maximum possible number of $H$-free orientations of an $n$-vertex graph. When $H$ is a tournament, the answer was determined precisely…

For positive integers $n>k>t$ let $\binom{[n]}{k}$ denote the collection of all $k$-subsets of the standard $n$-element set $[n]=\{1,\ldots,n\}$. Subsets of $\binom{[n]}{k}$ are called $k$-graphs. A $k$-graph $\mathcal{F}$ is called…