Related papers: Erd\H{o}s and Renyi conjecture

An old conjecture of Erd\H{o}s and McKay states that if all homogeneous sets in an $n$-vertex graph are of order $O(\log n)$ then the graph contains induced subgraphs of each size from $\{0,1,\ldots, \Omega (n^2)\}$. We prove a bipartite…

We prove for every graph H there exists a>0 such that, for every graph G with at least two vertices, if no induced subgraph of G is a subdivision of H, then either some vertex of G has at least a|G| neighbours, or there are two disjoint…

An n-vertex graph is called C-Ramsey if it has no clique or independent set of size C log n. All known constructions of Ramsey graphs involve randomness in an essential way, and there is an ongoing line of research towards showing that in…

For given graphs G1 and G2 the Ramsey number R(G1,G2), is the smallest positive integer n such that each blue-red edge coloring of the complete graph Kn contains a blue copy of G1 or a red copy of G2. In 1983, Erdos conjectured that there…

In their celebrated paper [Ramsey-Type Theorems, Discrete Appl. Math. 25 (1989) 37-52], Erd\H{o}s and Hajnal asked the following: is it true, that for any finite graph H there exists a constant c(H) such that for any finite graph G, if G…

Erd\H{o}s and Hajnal conjectured that, for every graph $H$, there exists a constant $c_H$ such that every graph $G$ on $n$ vertices which does not contain any induced copy of $H$ has a clique or a stable set of size $n^{c_H}$. We prove that…

A folklore result attributed to P\'olya states that there are $(1 + o(1))2^{\binom{n}{2}}/n!$ non-isomorphic graphs on $n$ vertices. Given two graphs $G$ and $H$, we say that $G$ is a unique subgraph of $H$ if $H$ contains exactly one…

We prove that for every integer $k$, there exists $\varepsilon > 0$ such that for every n-vertex graph $G$ with no pivot-minor isomorphic to $C_k$, there exist disjoint sets $A,B \subseteq V(G)$ such that $|A|,|B| \geq \varepsilon n$, and…

A vertex of a graph is bisimplicial if the set of its neighbors is the union of two cliques; a graph is quasi-line if every vertex is bisimplicial. A recent result of Chudnovsky and Seymour asserts that every non-empty even-hole-free graph…

A result due to Gy\'arf\'as, Hubenko, and Solymosi (answering a question of Erd\"os) states that if a graph $G$ on $n$ vertices does not contain $K_{2,2}$ as an induced subgraph yet has at least $c\binom{n}{2}$ edges, then $G$ has a…

We prove that for all $r\geq2$ and c>0, every graph of order n with at least cn^{r} cliques of order r contains a complete r-partite graph with each part of size $\lfloor c^{r}\log n \rfloor.$ This result implies a concise form of the…

Let $c$ denote the largest constant such that every $C_{6}$-free graph $G$ contains a bipartite and $C_4$-free subgraph having $c$ fraction of edges of $G$. Gy\H{o}ri et al. showed that $\frac{3}{8} \le c \le \frac{2}{5}$. We prove that…

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…

The Ramsey number $r(G)$ of a graph $G$ is the minimum $N$ such that every red-blue coloring of the edges of the complete graph on $N$ vertices contains a monochromatic copy of $G$. Determining or estimating these numbers is one of the…

We prove that for every k, there exists $c_k>0$ such that every graph G on n vertices not inducing a path $P_k$ and its complement contains a clique or a stable set of size $n^{c_k}$.

We prove the following 30-year old conjecture of Gy\H{o}ri and Tuza: the edges of every $n$-vertex graph $G$ can be decomposed into complete graphs $C_1,\ldots,C_\ell$ of orders two and three such that $|C_1|+\cdots+|C_\ell|\le…

In \cite{reed97}, Reed conjectures that the inequality $\chi (G) \leq \left \lceil \textstyle {1/2} (\omega (G) + \Delta (G) + 1) \right \rceil$ holds for any graph $G$. We prove this holds for a graph $G$ if $\bar{G}$ is disconnected. From…

One of the most basic questions one can ask about a graph $H$ is: how many $H$-free graphs on $n$ vertices are there? For non-bipartite $H$, the answer to this question has been well-understood since 1986, when Erd\H{o}s, Frankl and R\"odl…

In 1977, Erd\H{o}s and Hajnal made the conjecture that, for every graph $H$, there exists $c>0$ such that every $H$-free graph $G$ has a clique or stable set of size at least $|G|^c$; and they proved that this is true with $ |G|^c$ replaced…

For c in [0,1] let P_n(c) denote the set of n-vertex perfect graphs with density c and C_n(c) the set of n-vertex graphs without induced C_5 and with density c. We show that log|P_n(c)|/binom{n}{2}=log|C_n(c)|/binom{n}{2}=h(c)+o(1) with…