相关论文: A clique version of the Erd\H{o}s-Gallai stability…

The Erd\H{o}s-Gallai Theorem states that for $k \geq 2$, every graph of average degree more than $k - 2$ contains a $k$-vertex path. This result is a consequence of a stronger result of Kopylov: if $k$ is odd, $k=2t+1\geq 5$, $n \geq…

The famous Erd\H{o}s-Gallai Theorem on the Tur\'an number of paths states that every graph with $n$ vertices and $m$ edges contains a path with at least $\frac{2m}{n}$ edges. In this note, we first establish a simple but novel extension of…

In this note, we establish Andr\'{a}sfai--Erd\H{o}s--S\'{o}s-type stability theorems for two generalized Tur\'{a}n problems involving odd cycles, both of which are extensions of the Erd\H{o}s Pentagon Problem. Our results strengthen…

The Erd\H{o}s--Gallai Theorem states that for $k\geq 3$ every graph on $n$ vertices with more than $\frac{1}{2}(k-1)(n-1)$ edges contains a cycle of length at least $k$. Kopylov proved a strengthening of this result for 2-connected graphs…

A classic theorem of Erd\H{o}s and P\'osa (1965) states that every graph has either $k$ vertex-disjoint cycles or a set of $O(k \log k)$ vertices meeting all its cycles. While the standard proof revolves around finding a large `frame' in…

The Erd\H{o}s-Gallai Theorem states that every graph of average degree more than $l-2$ contains a path of order $l$ for $l\ge 2$. In this paper, we obtain a stability version of the Erd\H{o}s-Gallai Theorem in terms of minimum degree. Let…

The celebrated Erd\H{o}s-Ko-Rado (EKR) theorem for Paley graphs (of square order) states that all maximum cliques are canonical in the sense that each maximum clique arises from the subfield construction. Recently, Asgarli and Yip extended…

In this paper, we extend and refine previous Tur\'an-type results on graphs with a given circumference. Let $W_{n,k,c}$ be the graph obtained from a clique $K_{c-k+1}$ by adding $n-(c-k+1)$ isolated vertices each joined to the same $k$…

We generalize several classical theorems in extremal combinatorics by replacing a global constraint with an inequality which holds for all objects in a given class. In particular we obtain generalizations of Tur\'an's theorem, the…

The Erd\H{o}s--Gallai Theorem states that for $k \geq 3$, any $n$-vertex graph with no cycle of length at least $k$ has at most $\frac{1}{2}(k-1)(n-1)$ edges. A stronger version of the Erd\H{o}s--Gallai Theorem was given by Kopylov: If $G$…

For a simple graph $G$, let $n$ and $m$ denote the number of vertices and edges in $G$, respectively. The Erd\H{o}s-Gallai theorem for paths states that in a simple $P_k$-free graph, $m \leq \frac{n(k-1)}{2}$, where $P_k$ denotes a path…

A key feature of Simonovits' proof of the classic Erd\H{o}s-P\'osa theorem is a simple subgraph of the host graph, a frame, that determines the outcome of the theorem. We transfer this frame technique to $A$-paths. With it we deduce a…

We prove that a family of pseudo-Paley graphs of square order obtained from unions of cyclotomic classes satisfies the Erd\H{o}s-Ko-Rado (EKR) module property, in a sense that the characteristic vector of each maximum clique is a linear…

Malec and Tompkins (EUJC, 2023) considered the localized versions of Tur\'an-type problems, and proved a localized theorem on Erd\H{o}s-Gallai Theorem on paths. Zhao and Zhang (JGT, 2025) gave a long proof of a localized version of…

A classic result of Erd\H{o}s and P\'osa says that any graph contains either $k$ vertex-disjoint cycles or can be made acyclic by deleting at most $O(k \log k)$ vertices. Here we generalize this result by showing that for all numbers $k$…

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…

We prove a general stability theorem for $p$-class groups of number fields along relative cyclic extensions of degree $p^2$, which is a generalization of a finite-extension version of Fukuda's theorem by Li, Ouyang, Xu and Zhang. As an…

If $\Gamma$ is a graph for which every edge is in exactly one clique of order $\omega$, then one can form a new graph with vertex set equal to these cliques. This is a generalization of the line graph of $\Gamma$. We discover many general…

In 1965, Erd\H{o}s and P\'{o}sa proved that there is an (approximate) duality between the maximum size of a packing of cycles and the minimum size of a vertex set hitting all cycles. Such a duality does not hold for odd cycles, and Dejter…

We revisit the construction of Castella and Do of an anticyclotomic Euler system for the $p$-adic Galois representation of a modular form, using diagonal classes. Combining this construction and some previous results of ours, we obtain new…