On a problem of Erdős concerning primitive sequences

On an Extremal Problem concerning Primitive Sequences

A sequence a,< . . . of integers is called primitive if no a divides any other . (a 1 < . . . will always denote a primitive sequence .) It is easy to see that if a i < . . . < a,,, < n then max k = [(n + 1) /2] . The following question seems to be very much more difficult . Put f(n) =niax E 1) , a,, where the maximum is taken over all primitive sequences all of wliose terms are not exceeding n...

On a Problem of Steinhaus Concerning Binary Sequences

A small step forwards on the Erdős-Sós problem concerning the Ramsey numbers R(3, k)

Let ∆s = R(K3,Ks) − R(K3,Ks−1), where R(G,H) is the Ramsey number of graphs G and H defined as the smallest n such that any edge coloring of Kn with two colors contains G in the first color or H in the second color. In 1980, Erdős and Sós posed some questions about the growth of ∆s. The best known concrete bounds on ∆s are 3 ≤ ∆s ≤ s, and they have not been improved since the stating of the pro...

On the Erdős Discrepancy Problem

According to the Erdős discrepancy conjecture, for any infinite ±1 sequence, there exists a homogeneous arithmetic progression of unbounded discrepancy. In other words, for any ±1 sequence (x1, x2, ...) and a discrepancy C, there exist integers m and d such that | ∑ m i=1 xi·d| > C. This is an 80-year-old open problem and recent development proved that this conjecture is true for discrepancies ...

Reflections on a Problem of Erdős and Hajnal

The problem I am going to comment reached me in 1987 at Memphis in a letter of Uncle Paul. He wrote: 'We have the following problem with Hajnal. If G(n) has n points and does not contain induced C4, is it true that it has either a clique or an independent set with n 10 points? Kind regards to your boss+ colleagues, kisses to the E -s.' E.P.' After noting that E have been used in different conte...