Related papers: The Graham conjecture implies the Erdos-Turan conj…

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…

In 1965 Erd\H os conjectured that for all $k\ge2$, $s\ge1$ and $n\ge k(s+1)$, an $n$-vertex $k$-uniform hypergraph $\F$ with $\nu(\F)=s$ cannot have more than \newline $\max\{\binom{sk+k-1}k,\;\binom nk-\binom{n-s}k\}$ edges. It took almost…

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 $H$,…

An old question of Erdos asks if there exists, for each number N, a finite set S of integers greater than N and residue classes r(n) mod n for n in S whose union is all the integers. We prove that if $\sum_{n\in S} 1/n$ is bounded for such…

The basic theme of this paper is the fact that if $A$ is a finite set of integers, then the sum and product sets cannot both be small. A precise formulation of this fact is Conjecture 1 below due to Erd\H os-Szemer\'edi [E-S]. (see also…

The extremal number $\mathrm{ex}(n,F)$ of a graph $F$ is the maximum number of edges in an $n$-vertex graph not containing $F$ as a subgraph. A real number $r \in [1,2]$ is realisable if there exists a graph $F$ with $\mathrm{ex}(n , F) =…

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…

In [8], the gradient conjecture of R. Thom was proven for gradient flows of analytic functions on Rn. This result means that the secant at a limit point converges, so that the flow cannot spiral forever. Once the trajectory becomes…

The famous Tetrahedron Conjecture of Tur\'an from the 1940s asserts that the number of edges in an $n$-vertex $3$-graph without the tetrahedron, the complete $3$-graph on four vertices, cannot exceed that of the balanced complete cyclic…

In 1975 Szemer\'edi proved the long-standing conjecture of Erd\H{o}s and Tur\'an that any subset of $\bbZ$ having positive upper Banach density contains arbitrarily long arithmetic progressions. Szemer\'edi's proof was entirely…

A rough structure theorem is proved for graphs $G$ containing no copy of a bounded degree tree $T$: from any such $G$, one can delete $o(|G||T|)$ edges in order to get a subgraph all of whose connected components have a cover of order…

Seymour's Second-Neighborhood Conjecture states that every directed graph whose underlying graph is simple has at least one vertex $v$ such that the number of vertices of out-distance $2$ from $v$ is at least as large as the number of…

The famous Brown-Erd\H{o}s-S\'os conjecture from 1973 states, in an equivalent form, that for any fixed $\delta>0$ and integer $k\geq 3$ every sufficiently large linear $3$-uniform hypergraph of size $\delta n^2$ contains some $k$ edges…

We give a short proof of a sumset conjecture of Erd\"os, recently proved by Moreira, Richter and Robertson: every subset of the integers of positive density contains the sum of two infinite sets. The proof is written in the framework of…

A sequence $s_1,s_2,\ldots, s_k$ of elements of a group $G$ is called a valid ordering if the partial products $s_1, s_1 s_2, \ldots, s_1\cdots s_k$ are all distinct. A long-standing problem in combinatorial group theory asks whether, for a…

Let $G$ be a finite abelian group, and $r$ be a multiple of its exponent. The generalized Erd\H{o}s-Ginzburg-Ziv constant $s_r(G)$ is the smallest integer $s$ such that every sequence of length $s$ over $G$ has a zero-sum subsequence of…

This paper is an extension program of the notion of circle of partition developed in our first paper \cite{CoP}. As an application we prove the Erd\H{o}s-Tur\'{a}n additive base conjecture.

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.…

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…

We study the Tur\'{a}n problem for highly symmetric bipartite graphs arising from geometric shapes and periodic tilings commonly found in nature. 1. The prism $C_{2\ell}^{\square}:=C_{2\ell}\square K_{2}$ is the graph consisting of two…