On Rado's Boundedness Conjecture

On Rado's Boundedness Conjecture

We prove that Rado’s Boundedness Conjecture from Richard Rado’s 1933 famous dissertation Studien zur Kombinatorik is true if it is true for homogeneous equations. We then prove the first nontrivial case of Rado’s Boundedness Conjecture: if a1, a2, and a3 are integers, and if for every 24-coloring of the positive integers (or even the nonzero rational numbers) there is a monochromatic solution t...

$L^p$-Conjecture on Hypergroups

In this paper, we study $L^p$-conjecture on locally compact hypergroups and by some technical proofs we give some sufficient and necessary conditions  for a weighted Lebesgue space  $L^p(K,w)$ to be a convolution Banach algebra, where $1<p<infty$, $K$ is a locally compact hypergroup and $w$ is a weight function on $K$.  Among the other things, we also show that if $K$ is a locally compact hyper...

On Robson's Convergence and Boundedness Conjecture Concering the Height of Binary Search Trees

On some generalisations of Brown's conjecture

Let $P$ be a complex polynomial of the form $P(z)=zdisplaystyleprod_{k=1}^{n-1}(z-z_{k})$,where $|z_k|ge 1,1le kle n-1$ then $ P^prime(z)ne 0$. If $|z|

Variations on Ω - Boundedness

Let P be a property (or, equivalently, a class) of topological spaces. A space X is called P-bounded if every subspace of X with (or in) P has compact closure. Thus, countable-bounded has been known as ω-bounded and (σ-compact)-bounded as strongly ω-bounded. In this paper we present a systematic study of the interrelations of these two known “boundedness” concepts with P-boundedness where P is ...