Lower bounds for Z-numbers

Lower bounds for Z-numbers

Let p/q be a rational noninteger number with p > q ≥ 2. A real number λ > 0 is a Zp/q-number if {λ(p/q)n} < 1/q for every nonnegative integer n, where {x} denotes the fractional part of x. We develop several algorithms to search for Zp/q-numbers, and use them to determine lower bounds on such numbers for several p and q. It is shown, for instance, that if there is a Z3/2-number, then it is grea...

New Lower Bounds for Heilbronn Numbers

The n-th Heilbronn number, Hn, is the largest value such that n points can be placed in the unit square in such a way that all possible triangles defined by any three of the points have area at least Hn. In this note we establish new bounds for the first Heilbronn numbers. These new values have been found by using a simple implementation of simulated annealing to obtain a first approximation an...

New lower bounds for hypergraph Ramsey numbers

The Ramsey number rk(s, n) is the minimum N such that for every red-blue coloring of the k-tuples of {1, . . . , N}, there are s integers such that every k-tuple among them is red, or n integers such that every k-tuple among them is blue. We prove the following new lower bounds for 4-uniform hypergraph Ramsey numbers: r4(5, n) > 2 n log n and r4(6, n) > 2 2 1/5 , where c is an absolute positive...

Lower Bounds for Measurable Chromatic Numbers

The Lovász theta function provides a lower bound for the chromatic number of finite graphs based on the solution of a semidefinite program. In this paper we generalize it so that it gives a lower bound for the measurable chromatic number of distance graphs on compact metric spaces. In particular we consider distance graphs on the unit sphere. There we transform the original infinite semidefinit...

Some Lower Bounds for Estrada Index