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Let lambda be a singular cardinal such that both lambda and mathrm cf ( lambda) are aleph 0 -inaccessible. Does lambda to ( lambda, aleph 1)^2 hold?
Assume the Generalised Continuum Hypothesis and let beta= omega omega+1 . Is it true that aleph beta+1 not to ( aleph beta,4, ldots,4)^3 (where there are countably infinite many copies of 4 )?
Let G be the graph with vertex set those pairs (x,y) in mathbb N ^2 with mathrm gcd (x,y)=1 , in which we join two vertices if the differ in only one coordinate, and there by pm 1 . Is there a path going to inf…
Let A subseteq [1,n) be a set of integers such that (a,b)=1 for all distinct a,b in A . Is it true that sum a in A frac 1 n-a leq sum p<n frac 1 p +O(1)?
Let A= a 1<a 2< cdots be a sequence of integers which tends to infinity sufficiently fast. If there is an n such that all n+a k are primes then must there exist infinitely many such n ? What if we ask for $n+a…
For d geq 2 let F d(n) be minimal such that every set of n points in mathbb R ^d contains a set of F d(n) points with distinct distances. Estimate F d(n) for fixed d as n to infty .
Let P d(n) be such that in any set of n points in mathbb R ^d there exist at least P d(n) many points which do not contain an isosceles triangle. Estimate P d(n) - in particular, is it true that P 2(n)<n^ …
Does 1,2^3, ldots,N^3 contain a Sidon set of size gg N ? Is there an infinite set A subset mathbb N of positive density such that a^3 : a in A is a Sidon set?
We call a sequence of integers 0 leq a 1< cdots <a k admissible if it is missing at least one congruence class modulo every prime p . Let A(k)= min a k . Estimate A(k) - in particular, is it true that A(k) sim …
If omega(n) counts the number of distinct prime divisors of n then let F(n)= max k omega(n+k) frac log log k log k . Prove that F(n) to infty as n to infty .
Is it true that for every epsilon, eta 0 there exists a k such that the density of n for which P(n(n+1) cdots(n+k)) n^ 1- epsilon is at least 1- eta (where P(m) is the greatest prime divisor of m )?
There exists a constant C such that for all large x there is a collection of primes p 1< ldots<p k<x with sum frac 1 p i <C together with a system of congruences a i pmod p i such that every integer n<x sa…
Is it true that in any 2 -colouring of mathbb N there exists an infinite set A such that all elements of A+A are the same colour?
Let A subset mathbb N be such that every integer n geq 1 can be written uniquely as a n-b n for some a n,b n in A . How fast must a n/n increase?
For A subset mathbb N let f r(n) count the number of solutions to n=a 1+ cdots+a r with a i in A . Does there exist, for all r geq 2 , a basis A of order r (so that f r(n) 0 for all large n ) such that …
Let A subset mathbb N be an infinite Sidon set. Is it true that liminf x to infty frac lvert A cap [1,x] rvert x^ 1/2 ( log x)^ 1/2 =0? Does there exist an infinite Sidon set A such that liminf …
Call a set of distinct integers 1<n 1< cdots<n k a covering set if there is a choice of a i pmod n i for 1 leq i leq k such that every integer satisfies at least one of these congruences. A set is an irreducible c…
Call a set of distinct integers 1<n 1< cdots<n k with associated congruence classes a i pmod n i a distinct covering system if every integer satisfies at least one of these congruences. A minimal distinct covering s…
Let delta k be such that in any 2 -colouring of 1, ldots,n there exist at least ( delta k+o(1))n^2 many monochromatic k -term arithmetic progressions. Give reasonable bounds (or even an asymptotic formula) f…
Let f(n,k) count the number of 1 leq i leq k such that P(n+i) k (where P(m) is the largest prime divisor of m ). Is it true that, if alpha 1 is such that n=k^ alpha+o(1) , then f(n,k)=(1- rho( alpha)+o…
Let f(n) be maximal such that in any 2 -colouring of the subsets of 1, ldots,n there is always a monochromatic family of at least f(n) sets which is closed under taking unions and intersections. Estimate $f(n)…
Let f(n) be maximal such that there is a connected graph G with n vertices and f(n) edges such that R(K 3,G)= 2n-1. Let F(n) be maximal such that every connected graph G with n vertices and $ leq F(…
Let q(n,k) denote the least prime which does not divide prod 1 leq i leq k (n+i) . Is it true that there exists some c 0 such that, for all large n , q(n, log n)<(1-c)( log n)^2?
For r geq 3 let d r(e) be the minimal d such that mathrm ex r(n, mathcal F )=o(n^2), where mathcal F is the family of r -uniform hypergraphs on d vertices with e edges. Prove that d r(e)=(r-…
Let G be a finite 3 -uniform hypergraph, and let F G( kappa) denote the collection of 3 -uniform hypergraphs with chromatic number kappa not containing G . If F G( aleph 1) is not empty then there exists $X …