Abstract:
ErdÖs asks if it is possible to have n points in general position in the plane (no three on a line or four on a circle) such that for every i (1≤i≤n-1 ) there is a distance determined by the points that occur exactly i times. So far some examples have been discovered for 2≤n≤8 [1] [2]. A solution for the 8 point is provided by I. Palasti [3]. Here two other possible solutions for the 8 point case as well as all possible answers to 4 - 7 point cases are provided and finally a brief discussion on the generalization of the problem to higher dimensions is given.

Abstract:
A rational set in the plane is a point set with all its pairwise distances rational. Ulam and Erd\'os conjectured in 1945 that there is no dense rational set in the plane. In this paper we associate special surfaces, we call them distance surfaces, to finite (rational) sets in the plane. We prove that under a mild condition on the points in the rational set $S$ the associated distance surface in $\mathbb{P}^3$ is a surface of general type. Also if we assume Bombieri-Lang Conjecture in arithmetic algebraic geometry we can answer the Erd\'os-Ulam problem.

Abstract:
Affirming a conjecture of Erd\H{o}s and Renyi we prove that for any (real number) c_1>0 for some c_2>0, if a graph G has no c_1(log n) nodes on which the graph is complete or edgeless (i.e. G exemplifies |G| not-> (c_1 log n)^2_2) then G has at least 2^{c_2n} non-isomorphic (induced) subgraphs.

Abstract:
Erd\H{o}s conjectured that for any set $A\subseteq \mathbb{N}$ with positive lower asymptotic density, there are infinite sets $B,C\subseteq \mathbb{N}$ such that $B+C\subseteq A$. We verify Erd\H{o}s' conjecture in the case that $A$ has Banach density exceeding $\frac{1}{2}$. As a consequence, we prove that, for $A\subseteq \mathbb{N}$ with positive Banach density (a much weaker assumption than positive lower density), we can find infinite $B,C\subseteq \mathbb{N}$ such that $B+C$ is contained in the union of $A$ and a translate of $A$. Both of the aforementioned results are generalized to arbitrary countable amenable groups. We also provide a positive solution to Erd\H{o}s' conjecture for subsets of the natural numbers that are pseudorandom.

Abstract:
We investigate a conjecture of Paul Erd s, the last unsolved problem among those proposed in his landmark paper [2]. The conjecture states that there exists an absolute constant $C > 0$ such that, if $v_1, dots, v_n$ are unit vectors in a Hilbert space, then at least $C frac{2n}{n}$ of all $epsilon in {-1,1}^n$ are such that $|sum_{i=1}^n epsilon_i v_i| leq 1$. We disprove the conjecture. For Hilbert spaces of dimension $d > 2,$ the counterexample is quite strong, and implies that a substantial weakening of the conjecture is necessary. However, for $d = 2,$ only a minor modification is necessary, and it seems to us that it remains a hard problem, worthy of Erd s. We prove some weaker related results that shed some light on the hardness of the problem.

Abstract:
Let $ES(n)$ denote the minimum natural number such that every set of $ES(n)$ points in general position in the plane contains $n$ points in convex position. In 1935, Erd\H{o}s and Szekeres proved that $ES(n) \le {2n-4 \choose n-2}+1$. 26 years later, they obtained the lower bound $2^{n-2}+1 \le ES(n)$. In a recent paper, Vlachos proved that $\limsup\limits_{n\rightarrow\infty} \frac{ES(n)}{{2n-5 \choose n-2}} \le \frac{29}{32}$. In this paper, we mostly use the ideas and tools from Vlachos' paper and improve the bound to $\limsup\limits_{n\rightarrow\infty} \frac{ES(n)}{{2n-5 \choose n-2}} \le \frac{7}{8}$.

Abstract:
Let f(n) denote the smallest positive integer such that every set of $f(n)$ points in general position in the Euclidean plane contains a convex n-gon. In a seminal paper published in 1935, Erd\H{o}s and Szekeres proved that f(n) exists and provided an upper bound. In 1961, they also proved a lower bound, which they conjectured is optimal. Their bounds are: $2^{n-2}+1 \leq f(n) \leq {2n - 4 \choose n-2}+1$. Since then, the upper bound has been improved by rougly a factor of 2, to $f(n) \leq {2n - 5 \choose n-2}+1$. In the current paper, we further improve the upper bound by proving that: $$ \limsup\limits_{n\rightarrow \infty} \frac{f(n)}{{2n-5 \choose n-2}} \leq \frac{29}{32}$$

Abstract:
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 central problems in combinatorics. One of the oldest results in Ramsey Theory, proved by Erd\H{o}s and Szekeres in 1935, asserts that the Ramsey number of the complete graph with $m$ edges is at most $2^{O(\sqrt{m})}$. Motivated by this estimate Erd\H{o}s conjectured, more than a quarter century ago, that there is an absolute constant $c$ such that $r(G) \leq 2^{c\sqrt{m}}$ for any graph $G$ with $m$ edges and no isolated vertices. In this short note we prove this conjecture.

Abstract:
In this paper analyzes \textit{The Erd\H{o}s-Straus conjecture} asserts that $f$$(n)$ $>$ 0 for every $n$ $\geq$ 2, where $f(n)$ indicates the number of solutions to the Diophantine Equation $\frac{4}{n}=\frac{1}{n_1}+\frac{1}{n_2}+\frac{1}{n_3}$. We show that there exists a function $G(p)$ to be a boundary asymptotic of $\sum_{p\leq{N}}f_{I}(p)$, which will have an associated error. We analyze the case when n is a prime number, this was separately developed by Terence Tao [8] and Jia [1], [2].