6
votes
1answer
333 views

Bounds on the size of sets not containing a given finite pattern

Recall the following version of Szemerédi's Theorem: let $r_k(N)$ be the largest cardinality of a subset of $[N]:=\{1,\ldots, N\}$ which does not contain an arithmetic progression of length $k$. Then, ...
12
votes
2answers
579 views

Arithmetic progressions modulo $p$ under the squaring map

I feel that the following problem should be known, but I'm not sure where to look for it. Fix a real constant $\frac{1}{2} \ge \epsilon > 0$. For varying primes $p$, Let $A_p$ denote the set of ...
11
votes
3answers
730 views

What does the computer suggest about the parity of p(n), for n in a fixed arithmetic progression?

Let p(n) be the number of partitions of n. A famous theorem of Euler allows one to compute the parity of p(n) quickly for quite large n. In: On the distribution of parity in the partition function, ...
3
votes
1answer
677 views

Can every finite graph be represented by an arithmetic sequence of natural numbers?

(This is a follow-up to my previous questions Natural models of graphs?.) Erdös in The Representation of a Graph by Set Intersections (1966) states: Theorem. Let $G$ be an arbitrary graph. Then ...