相关论文: Tur\'an-Type Extremal Results for Distance-$k$ Gra…

For two graphs $G$ and $H$, the Tur\'{a}n number $ex(G,H)$ is the maximum number of edges in a subgraph of $G$ that contains no copy of $H$. Chen, Li, and Tu determined the Tur\'{a}n numbers $ex(K_{m,n},kK_2)$ for all $k\geq 1$ [7]. In this…

Given a measurable set $A\subset \mathbb R^d$ we consider the "large-distance graph" $\mathcal{G}_A$, on the ground set $A$, in which each pair of points from $A$ whose distance is bigger than 2 forms an edge. We consider the problems of…

In a generalized Tur\'an problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free…

We prove that the number of multigraphs with vertex set $\{1, \ldots, n\}$ such that every four vertices span at most nine edges is $a^{n^2 + o(n^2)}$ where $a$ is transcendental (assuming Schanuel's conjecture from number theory). This is…

Generalized Tur\'an problems have been a central topic of study in extremal combinatorics throughout the last few decades. One such problem is maximizing the number of cliques of size $t$ in a graph of a fixed order that does not contain…

As a variant of the famous Tur\'an problem, we study $\mathrm{rex}(n,F)$, the maximum number of edges that an $n$-vertex regular graph can have without containing a copy of $F$. We determine $\mathrm{rex}(n,K_{r+1})$ for all pairs of…

Generalizing Tur\'an's classical extremal problem, Alon and Shikhelman investigated the problem of maximizing the number of $T$ copies in an $H$-free graph, for a pair of graphs $T$ and $H$. Whereas Alon and Shikhelman were primarily…

The Tur\'an number $\mathrm{ex}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex graph which does not contain $H$ as a subgraph. The Tur\'{a}n number of regular polyhedrons was widely studied in a series of works due to…

Given two graphs $H$ and $F$, the generalized Tur\'an number $\mathrm{ex}(n,H,F)$ is the largest number of copies of $H$ in an $n$-vertex $F$-free graph. For every $F$ and sufficiently large $n$, we present an extremal graph for a…

A fundamental barrier in extremal hypergraph theory is the presence of many near-extremal constructions with very different structures. Indeed, the classical constructions due to Kostochka imply that the notorious extremal problem for the…

Given a family of graphs $\mathcal{F}$, the Tur\'{a}n number $ex(n, \mathcal{F})$ denotes the maximum number of edges in any $\mathcal{F}$-free graph on $n$ vertices. Recently, Alon and Frankl studied of maximum number of edges in an…

Let $C_{\ell}$ be the cycle of order ${\ell}$. The square of $C_{\ell}$, denoted by $C_{\ell}^2$, is obtained by joining all pairs of vertices with distance no more than two in $C_{\ell}$. A graph is called $F$-free if it does not contain…

For a graph $H$, the {\em extremal number} $ex(n,H)$ is the maximum number of edges in a graph of order $n$ not containing a subgraph isomorphic to $H$. Let $\delta(H)>0$ and $\Delta(H)$ denote the minimum degree and maximum degree of $H$,…

The fundamental theorem of Tur\'{a}n from Extremal Graph Theory determines the exact bound on the number of edges $t_r(n)$ in an $n$-vertex graph that does not contain a clique of size $r+1$. We establish an interesting link between…

Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting $P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all sets $P_i \cup P_j$ with $i \neq j$. This can be thought of as the `$k$-expansion' of the complete…

The Tur\'{a}n number of a graph $H$ is the maximum number of edges in any graph of order $n$ that does not contain $H$ as a subgraph. In 1959, Erd\H os and Gallai obtained a sharp upper bound of Tur\'{a}n numbers for a path of arbitrary…

For two graphs $G$ and $F$, the extremal number of $F$ in $G$, denoted by {ex}$(G,F)$, is the maximum number of edges in a spanning subgraph of $G$ not containing $F$ as a subgraph. Determining {ex}$(K_n,F)$ for a given graph $F$ is a…

Very recently, Alon and Frankl initiated the study of the maximum number of edges in $n$-vertex $F$-free graphs with matching number at most $s$. For fixed $F$ and $s$, we determine this number apart from a constant additive term. We also…

We determine the maximum possible number of edges of a graph with $n$ vertices, matching number at most $s$ and clique number at most $k$ for all admissible values of the parameters.

Tur\'{a}n type extremal problem is how to maximize the number of edges over all graphs which do not contain fixed forbidden subgraphs. Similarly, spectral Tur\'{a}n type extremal problem is how to maximize (signless Laplacian) spectral…