Cross-intersecting families and primitivity of symmetric systems

Let $X$ be a finite set and $\mathfrak p\subseteq 2^X$, the power set of $X$, satisfying three conditions: (a) $\mathfrak p$ is an ideal in $2^X$, that is, if $A\in \mathfrak p$ and $B\subset A$, then $B\in \mathfrak p$; (b) For $A\in 2^X$ with $|A|\geq 2$, $A\in \mathfrak p$ if $\{x,y\}\in \mathfrak p$ for any $x,y\in A$ with $x\neq y$; (c) $\{x\}\in \mathfrak p$ for every $x\in X$. The pair $(X,\mathfrak p)$ is called a symmetric system if there is a group $\Gamma$ transitively acting on $X$ and preserving the ideal $\mathfrak p$. A family $\{A_1,A_2,\ldots,A_m\}\subseteq 2^X$ is said to be a cross-$\mathfrak{p}$-family of $X$ if $\{a, b\}\in \mathfrak{p}$ for any $a\in A_i$ and $b\in A_j$ with $i\neq j$. We prove that if $(X,\mathfrak p)$ is a symmetric system and $\{A_1,A_2,\ldots,A_m\}\subseteq 2^X$ is a cross-$\mathfrak{p}$-family of $X$, then $$\sum_{i=1}^m|{A}_i|\leq\left\{ \begin{array}{cl} |X| & \hbox{if $m\leq \frac{|X|}{\alpha(X,\, \mathfrak p)}$,} \\ m\, \alpha(X,\, \mathfrak p) & \hbox{if $m\geq \frac{|X|}{\alpha{(X,\, \mathfrak p)}}$,} \end{array}\right.$$ where $\alpha(X,\, \mathfrak p)=\max\{|A|:A\in\mathfrak p\}$. This generalizes Hilton's theorem on cross-intersecting families of finite sets, and provides analogs for cross-$t$-intersecting families of finite sets, finite vector spaces and permutations, etc. Moreover, the primitivity of symmetric systems is introduced to characterize the optimal families.Comment: 15 page

Common Value Auctions with Return Policies

This paper examines the role of return policies in common value auctions. We first characterize the unique symmetric equilibrium in first-price and second-price auctions with continuous signals and discrete common values when certain return policies are provided. We then examine how the return policies affect a seller's revenue. When the lowest common value is zero, a more generous return policy generates a higher seller's revenue; the full refund policy extracts all the surplus and therefore implements the optimal selling mechanism; given any return policy, a second-price auction generates a higher revenue than a first-price auction. In a second-price auction where the lowest common value is not zero but still smaller than the seller's reservation value, then a more generous return policy also generates a higher revenue; otherwise, the optimal return policy could be a full refund, no refund or partial refund policy.auctions, return policies, refund

Charged Scalar Perturbations around Garfinkle-Horowitz-Strominger Black Holes

We examine the stability of the Garfinkle-Horowitz-Strominger (GHS) black hole under charged scalar perturbations. We find that different from the neutral scalar field perturbations, only two numerical methods, such as the continued fraction method and the asymptotic iteration method, can keep high efficiency and accuracy requirements in the frequency domain computations. The comparisons of the efficiency between these two methods have also been done. Employing the appropriate numerical method, we show that the GHS black hole is always stable against charged scalar perturbations. This is different from the result obtained in the de Sitter and Anti-de Sitter black holes. Furthermore we argue that in the GHS black hole background there is no amplification of the incident charged scalar wave to cause the superradiance, so that the superradiant instability cannot exist in this spacetime.Comment: 24 pages, 5 figure

Superradiant instability of Kerr-de Sitter black holes in scalar-tensor theory

We investigate in detail the mechanism of superradiance to render the instability of Kerr-de Sitter black holes in scalar-tensor gravity. Our results provide more clues to examine the scalar-tensor gravity in the astrophysical black holes in the universe with cosmological constant. We also discuss the spontaneous scalarization in the de Sitter background and find that this instability can also happen in the spherical de Sitter configuration in a special style.Comment: (v2)21 pages, 21 figures; Sec. V revised; This version has been accepted for publication by JHE