On a problem of Erdős and Graham

In this paper we provide bounds for the size of the solutions of the Diophantine equation

$$\begin{aligned} x(x+1)(x+2)(x+3)(x+m)(x+m+1)(x+m+2)(x+m+3)=y^2, \end{aligned}$$

where \(4\le m\in \mathbb {N}\) is a parameter. We also determine all integral solutions for \(1\le m\le 10^6.\)

     

On a bottleneck bipartition conjecture of Erdős

For a graphG, let (U,V)=max{e(U), e(V)} for a bipartition (U, V) ofV(G) withUV=V(G),UV=Ø. Define (G)=min_(U,V ){(U,V)}. Paul Erds conjectures . This paper verifies the conjecture and shows .This work was part of the author's Ph. D. thesis at the University of New Mexico. Research Partially supported by NSA Grant MDA904-92-H-3050.     

On a conjecture of Erdős and Simonovits: Even cycles

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let C _k denote a cycle of length k, and let C _k denote the set of cycles C _?, where 3≤?≤k and ? and k have the same parity. Erd?s and Simonovits conjectured that for any family F consisting of bipartite graphs there exists an odd integer k such that ex(n,F ∪ C _k ) ～ z(n,F) — here we write f(n) ～ g(n) for functions f,g: ? → ? if lim _n→∞ f(n)/g(n)=1. They proved this when F ={C ₄} by showing that ex(n,{C ₄;C ₅})～z(n,C ₄). In this paper, we extend this result by showing that if ?∈{2,3,5} and k>2? is odd, then ex(n,C _2? ∪{C _k }) ～ z(n,C _2? ). Furthermore, if k>2?+2 is odd, then for infinitely many n we show that the extremal C _2? ∪{C _k }-free graphs are bipartite incidence graphs of generalized polygons. We observe that this exact result does not hold for any odd k<2?, and furthermore the asymptotic result does not hold when (?,k) is (3, 3), (5, 3) or (5, 5). Our proofs make use of pseudorandomness properties of nearly extremal graphs that are of independent interest.     

On a modification of a problem of Bialostocki, Erd?s, and Lefmann

For positive integers m and r, one can easily show there exist integers N such that for every map Δ:{1,2,…,N}→{1,2,…,r} there exist 2m integers

x₁<?<x_m<y₁<?<y_m,     

A generalization of a conjecture due to Erd?s, Jacobson and Lehel

An r-graph is a loopless undirected graph in which no two vertices are joined by more than r edges. An r-complete graph on m+1 vertices, denoted by , is an r-graph on m+1 vertices in which each pair of vertices is joined by exactly r edges. A non-increasing sequence π=(d₁,d₂,…,d_n) of nonnegative integers is r-graphic if it is realizable by an r-graph on n vertices. Let be the smallest even integer such that each n-term r-graphic sequence with term sum of at least is realizable by an r-graph containing as a subgraph. In this paper, we determine the value of for sufficiently large n, which generalizes a conjecture due to Erd?s, Jacobson and Lehel.     

Asymptotic results for partitions (II) and a conjecture of Bateman and Erdös

Asymptotic results are obtained for p_A^(k)(n), the kth difference of the function p_A(n) which is the number of partitions of n into integers from A. Under certain restrictions on A it is shown that

$\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k+1)}}\text{(n)}\text{P}{}_{\mathrm{A}}{}^{\mathrm{(k)}}\text{(n)}\text{= O(n}{}^{\mathrm{?1/2}}\text{) (n→ ∫)}$

thereby verifying for these A a conjecture of Bateman and Erdös.     

On a paper of Erdös and Szekeres

Propositions 1.1–1.3 stated below contribute to results and certain problems considered in [E-S], on the behavior of products\(\Pi^n_1(1-z^{a_j}),1\leq{a_1}...\leq{a_n}\) integers. In the discussion below, {a₁,..., a_n} will be either a proportional subset of {1,..., n} or a set of large arithmetic diameter.     

On a problem of Erdős and Moser

A set A of vertices in an r-uniform hypergraph \(\mathcal H\) is covered in \(\mathcal H\) if there is some vertex \(u\not \in A\) such that every edge of the form \(\{u\}\cup B\), \(B\in A^{(r-1)}\) is in \(\mathcal H\). Erd?s and Moser (J Aust Math Soc 11:42–47, 1970) determined the minimum number of edges in a graph on n vertices such that every k-set is covered. We extend this result to r-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.     

On a problem of Erdős and Rado

We give some improved estimates for the digraph Ramsey numbersr(K _n ^* ,L _m ), the smallest numberp such that any digraph of orderp either has an independent set ofn vertices or contains a transitive tournament of orderm. By results of Baumgartner and of Erdős and Rado, this is equivalent to the following infinite partition problem: for an infinite cardinal κ and positive integersn andm, find the smallest numberp such that

On a problem of Erdös and Alladi

We give an estimate for the quantity {f(n):nx, p(n)y}, wherep(n) denotes the greatest prime factor ofn andf belongs to a certain class of multiplicative functions. As an application, we show that for the Moebius function, ({(n):nx, p(n)y}) ({1:nx, p(n)y})^–1 tends to zero, asx, uniformly iny2, and thus settle a conjecture of Erdös.Supported by a grant from the Deutsche Forschungsgesellschaft.     

On a Question of Erdős and Ulam

Ulam asked in 1945 if there is an everywhere dense rational set, i.e., 1 a point set in the plane with all its pairwise distances rational. Erdős conjectured that if a set S has a dense rational subset, then S should be very special. The only known types of examples of sets with dense (or even just infinite) rational subsets are lines and circles. In this paper we prove Erdős’ conjecture for algebraic curves by showing that no irreducible algebraic curve other than a line or a circle contains an infinite rational set.     

On a problem of Erdős

Let s≥2 be an integer. Denote by f ₁(s) the least integer so that every integer l>f ₁(s) is the sum of s distinct primes. Erd?s proved that f ₁(s)<p ₁+p ₂+?+p _s +Cslogs, where p _i is the ith prime and C is an absolute constant. In this paper, we prove that f ₁(s)=p ₁+p ₂+?+p _s +(1+o(1))slogs=p ₂+p ₃+?+p _s+1+o(slogs). This answers a question posed by P. Erd?s.     

Edge coloring of hypergraphs and a conjecture of ErdÖs,Faber, Lovász

Call a bypergraphsimple if for any pairu, v of distinct vertices, there is at most one edge incident to bothu andv, and there are no edges incident to exactly one vertex. A conjecture of Erds, Faber and Lovász is equivalent to the statement that the edges of any simple hypergraph onn vertices can be colored with at mostn colors. We present a simple proof that the edges of a simple hypergraph onn vertices can be colored with at most [1.5n-2 colors].This research was partially supported by N.S.F. grant No. MCS-8311422.     

A variation of a conjecture due to Erdös and Sós

Erdoes and Soes conjectured in 1963 that every graph G on n vertices with edge number e（G） 〉 1/2（k - 1）n contains every tree T with k edges as a subgraph. In this paper, we consider a variation of the above conjecture, that is, for n 〉 9/ 2k^2 ＋ 37/2＋ 14 and every graph G on n vertices with e（G） 〉 1/2 （k- 1）n, we prove that there exists a graph G＇ on n vertices having the same degree sequence as G and containing every tree T with k edges as a subgraph.     

Generalizations of Graham’s pebbling conjecture

We investigate generalizations of pebbling numbers and of Graham’s pebbling conjecture that $\pi \left(G\square H\right)\le \pi \left(G\right)\pi \left(H\right)$, where $\pi \left(G\right)$ is the pebbling number of the graph $G$. We develop new machinery to attack the conjecture, which is now twenty years old. We show that certain conjectures imply others that initially appear stronger. We also find counterexamples that shows that Sjöstrand’s theorem on cover pebbling does not apply if we allow the cost of transferring a pebble from one vertex to an adjacent vertex to depend on the weight of the edge and we describe an alternate pebbling number for which Graham’s conjecture is demonstrably false.