Related papers: Tur\'an theorems for unavoidable patterns

The edge blow-up of a graph $F$ is the graph obtained from replacing each edge in $F$ by a clique of the same size where the new vertices of the cliques are all different. In this article, we concern about the Tur\'an problem for the edge…

Given a graph $H$, the Tur\'an number $ex(n,H)$ is the largest number of edges in an $H$-free graph on $n$ vertices. We make progress on a recent conjecture of Conlon, Janzer, and Lee on the Tur\'an numbers of bipartite graphs, which in…

We study the Tur\'{a}n numbers of $3$-graphs avoiding $3$-graphs $F$ and $M_{s+1}^3$, a matching of size $s+1$. We disprove a conjecture of Gerbner, Tompkins, and Zhou [European Journal of Combinatorics, 2025, 127:104155] on…

Caro, Patk\'os, and Tuza initiated a systematic study of the bipartite Tur\'an number for trees, and in particular asked for the extremal number of edges in connected bipartite graphs with prescribed color-class sizes that contain no paths…

For an odd integer $k$, let $\mathcal{C}_k = \{C_3,C_5,...,C_k\}$ denote the family of all odd cycles of length at most $k$ and let $\mathcal{C}$ denote the family of all odd cycles. Erd\H{o}s and Simonovits \cite{ESi1} conjectured that for…

A conjecture of Erd\H{o}s from 1967 asserts that any graph on $n$ vertices which does not contain a fixed $r$-degenerate bipartite graph $F$ has at most $Cn^{2-1/r}$ edges, where $C$ is a constant depending only on $F$. We show that this…

We prove that for any $r\in \mathbb{N}$, there exists a constant $C_r$ such that the following is true. Let $\mathcal{F}=\{F_1,F_2,\dots\}$ be an infinite sequence of bipartite graphs such that $|V(F_i)|=i$ and $\Delta(F_i)\leq \Delta$ hold…

Given a graph $G$, its $2$-color Tur\'{a}n number $\mathrm{ex}^{(2)}(n,G)$ is the largest number of edges in an $n$-vertex graph whose edges can be colored with two colors avoiding a monochromatic copy of $G$. Let…

If we two-colour a circle, we can always find an inscribed triangle with angles $(\frac{\pi}{7},\frac{2\pi}{7},\frac{4\pi}{7})$ whose three vertices have the same colour. In fact, Bialostocki and Nielsen showed that it is enough to consider…

A new tur\'an-type problem on distances on graphs was introduced by Tyomkyn and Uzzell. In this paper, we focus on the case that the distance is two. We primely show that for any value of $n$, a graph on $n$ vertices without three vertices…

The Tur\'an problem asks for the largest number of edges in an $n$-vertex graph not containing a fixed forbidden subgraph $F$. We construct a new family of graphs not containing $K_{s,t}$, for $t= C^s$, with $\Omega(n^{2-1/s})$ edges…

We introduce and study a new type of Ramsey-Turan problems, a typical example of which is the following one: let c>0 and G be a graph of sufficiently large order n with minimum degree >3n/4. If the edges of G are colored in blue or red,…

An important question in extremal graph theory raised by Vera T. S\'os asks to determine for a given integer $t\ge 3$ and a given positive real number $\delta$ the asymptotically supremal edge density $f_t(\delta)$ that an $n$-vertex graph…

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 $H$,…

Fox--Grinshpun--Pach showed that every $3$-coloring of the complete graph on $n$ vertices without a rainbow triangle contains a clique of size $\Omega\left(n^{1/3}\log^2 n\right)$ which uses at most two colors, and this bound is tight up to…

We study an anti-Ramsey extension of the classical Corr\'{a}di--Hajnal Theorem: how many colors are needed to color the complete graph on $n$ vertices in order to guarantee a rainbow copy of $t K_{3}$, that is, $t$ vertex-disjoint…

For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \geq b_r(G)$ for any $G$, and Tur\'an's Theorem says that equality holds for complete graphs.…

Guided by the connections between hypergraphs and exterior algebras, we study Tur\'an and Ramsey type problems for alternating multilinear maps. This study lies at the intersection of combinatorics, group theory, and algebraic geometry, and…

Thomason [$\textit{Trans. Amer. Math. Soc.}$ 296.1 (1986)] proved that every sufficiently large tournament contains Hamilton paths and cycles with all possible orientations, except possibly the consistently oriented Hamilton cycle. This…

A $\rho$-mean coloring of a graph is a coloring of the edges such that the average number of colors incident with each vertex is at most $\rho$. For a graph $H$ and for $\rho \geq 1$, the {\em mean Ramsey-Tur\'an number} $RT(n,H,\rho-mean)$…