相关论文: An Erd\H{o}s-Stone type result for high-order spec…

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 $G$…

We prove an extension of the Regularity Lemma with vertex and edge weights which can be applied for a large class of graphs. The applications involve random graphs and a weighted version of the Erd\H{o}s-Stone theorem. We also provide means…

The celebrated Mantel's theorem states that any triangle-free graph on $n$ vertices contains at most $\left\lfloor n^2/4\right\rfloor$ edges. It is natural to ask how many triangles must exist in a graph with more than $\left\lfloor…

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…

Erd\H{o}s conjectured that every $n$-vertex triangle-free graph contains a subset of $\lfloor n/2\rfloor$ vertices that spans at most $n^2/50$ edges. Extending a recent result of Norin and Yepremyan, we confirm this conjecture for graphs…

Given a graph $H$, a graph is $H$-free if it does not contain $H$ as a subgraph. We continue to study the topic of "extremal" planar graphs, that is, how many edges can an $H$-free planar graph on $n$ vertices have? We define…

The Erd\H{o}s-S\'os Conjecture states that every graph with average degree exceeding $k-1$ contains every tree with $k$ edges as a subgraph. We prove that there are $\delta>0$ and $k_0\in\mathbb N$ such that the conjecture holds for every…

A classical theorem of De Bruijn and Erd\H{o}s asserts that any noncollinear set of n points in the plane determines at least n distinct lines. We prove that an analogue of this theorem holds for graphs. Restricting our attention to…

For $0 \leq \alpha < 1$, the $\alpha$-spectral radius of a graph $G$ is defined as the largest eigenvalue of $A_{\alpha}(G)=\alpha D(G)+(1-\alpha)A(G)$, where $D(G)$ and $A(G)$ are the diagonal matrix of degrees and adjacency matrix of $G$,…

We obtain some new upper bounds on the maximum number $f(n)$ of edges in $n$-vertex graphs without containing cycles of length four. This leads to an asymptotically optimal bound on $f(n)$ for a broad range of integers $n$ as well as a…

In the 1960s, Erd\H{o}s and his cooperators initiated the research of the maximum numbers of edges in a graph or a planar graph on $n$ vertices without $k$ edge-disjoint cycles. This problem had been solved for $k\leq4$. As pointed out by…

Given an $(r + 1)$-chromatic graph $H$, the fundamental edge stability result of Erd\H{o}s and Simonovits says that all $n$-vertex $H$-free graphs have at most $(1 - 1/r + o(1)) \binom{n}{2}$ edges, and any $H$-free graph with that many…

This paper focuses on extensions of the classic Erd\H{o}s-Gallai Theorem for the set of weighted function of each edge in a graph. The weighted function of an edge $e$ of an $n$-vertex uniform hypergraph $\mathcal{H}$ is defined to a…

A well-known theorem of Erd\H{o}s and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Gy\H{o}ri, Katona and Lemons gave an extension of this result to hypergraphs by determining…

An extremal graph for a graph $H$ on $n$ vertices is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $T_{n,r}$ be the Tur\'{a}n graph, which is the complete $r$-partite graph on $n$ vertices…

Given a graph $L$, the Tur\'an number $\textrm{ex}(n,L)$ is the maximum possible number of edges in an $n$-vertex $L$-free graph. The study of Tur\'an number of graphs is a central topic in extremal graph theory. Although the celebrated…

In 1962, Erd\H{o}s gave a sufficient condition for Hamilton cycles in terms of the vertex number, edge number, and minimum degree of graphs which generalized Ore's theorem. One year later, Moon and Moser gave an analogous result for…

A famous conjecture of Erd\H{o}s asserts that for $k\ge 3$, the maximum number of edges in an $n$-vertex $k$-uniform hypergraph without $s+1$ pairwise disjoint edges is $\max\{\binom{n}{k}-\binom{n-s}{k},\binom{sk+k-1}{k}\}$. This problem…

One of the cornerstones of extremal graph theory is a result of F\"uredi, later reproved and given due prominence by Alon, Krivelevich and Sudakov, saying that if $H$ is a bipartite graph with maximum degree $r$ on one side, then there is a…

The following sharpening of Tur\'an's theorem is proved. Let $T_{n,p}$ denote the complete $p$--partite graph of order $n$ having the maximum number of edges. If $G$ is an $n$-vertex $K_{p+1}$-free graph with $e(T_{n,p})-t$ edges then there…