Related papers: From a $(p,2)$-Theorem to a Tight $(p,q)$-Theorem

To a special type of grope embedded in 4-space, that we call an admissible grope, we associate a length function for each real number q at least 1. This gives rise to a family of pseudo-metrics d^q, refining the slice genus metric, on the…

The determinant lower bound of Lovasz, Spencer, and Vesztergombi [European Journal of Combinatorics, 1986] is a powerful general way to prove lower bounds on the hereditary discrepancy of a set system. In their paper, Lovasz, Spencer, and…

We investigate the product measures of intersection problems in extremal combinatorics. Invoking a recent result of He--Li--Wu--Zhang, we prove that for any $ n \geq t \geq 3$ and $ p_1, p_2 \in (0, \frac{1}{t+1})$, if $ \mathcal{F}_1,…

Let $(P, Q)$ be a pair of Jacobian polynomials. We can show that $ <P, Q>+l+2g(P)-2= 0= <P, [P,Q]>$, where $<f, g>$ is the intersection number of $f, g\in \CC[x, y]$ in the affine plane, $l$ is the number of branch at point at infinity and…

Let A_1,...,A_k be a collection of families of subsets of an n-element set. We say that this collection is cross-intersecting if for any i,j in [k] with i not equal to j, A in A_i and B in A_j implies that the intersection of A and B is…

We introduce an analytic family of twisted Fourier transforms $\left\{\mathcal{F}^{(x)}_p\right\}_{x\in \mathbb{R},p\in [1,2)}$ for non-Kac compact quantum groups and establish a sharpened form of the Hausdorff-Young inequality in the range…

Given positive integers $p,q$. For any integer $k\ge2$, an edge coloring of the complete $k$-graph $K_n^{(k)}$ is said to be a $(p,q)$-coloring if every copy of $K_p^{(k)}$ receives at least $q$ colors. The Erd\H{o}s-Gy\'{a}rf\'{a}s…

We introduce a new ``Winding Number Conjecture'' about maps from the $(d-1)$-skeleton of the $((d+1)(q-1))$-simplex into $\real^d$. This conjecture is equivalent to the Topological Tverberg Theorem. Furthermore, many statements about the…

Three lines are concurrent if they intersect at a single point. In this paper I prove that if $F$ is a bounded family of compact connected sets in the plane, such that every three sets in $F$ can be pierced by a single line, then there…

The Weak approximation theorem describes the closure of $G(Q)$ inside $G(Q_p)$ as well as inside $G(R)$ for $G$ an algebraic group over $Q$; the closure is always an open normal subgroup with finite abelian quotient, and is well understood…

It was conjectured by Flynn that there exists a constant $\kappa$ such that, for any integer $g \ge 2$, any $m \le \kappa g$, there exists a hyperelliptic curve of genus $g$ over $\mathbb Q$ with a rational $m$-torsion point on its…

Let k be a regular F_p-algebra, let A = k[x,y]/(x^b - y^a) be the coordinate ring of a planar cuspical curve, and let I = (x,y) be the ideal that defines the cusp point. We give a formula for the relative K-groups K_q(A,I) in terms of the…

Holroyd and Propp used Hall's marriage theorem to show that, given a probability distribution pi on a finite set S, there exists an infinite sequence s_1,s_2,... in S such that for all integers k >= 1 and all s in S, the number of i in…

Let $\mathcal{F}$ be a family of graphs, and let $p,r$ be nonnegative integers. The \textsc{$(p,r,\mathcal{F})$-Covering} problem asks whether for a graph $G$ and an integer $k$, there exists a set $D$ of at most $k$ vertices in $G$ such…

For positive integers $m$ and $n$, define $f(m,n)$ to be the smallest integer such that any subset $A$ of the $m \times n$ integer grid with $|A| \geq f(m,n)$ contains a rectangle; that is, there are $x\in [m]$ and $y \in [n]$ and…

H.J. Zassenhaus conjectured that any unit of finite order and augmentation $1$ in the integral group ring $\mathbb{Z}G$ of a finite group $G$ is conjugate in the rational group algebra $\mathbb{Q}G$ to an element of $G$. We prove the…

A family of $k$-subsets $A_1, A_2, ..., A_d$ on $[n]=\{1,2,..., n\}$ is called a $(d, c)$-cluster if the union $A_1\cup A_2 \cup ... \cup A_d$ contains at most $ck$ elements with $c<d$. Let $\mathcal{F}$ be a family of $k$-subsets of an…

Let $(X, d)$ be a compact metric space, and let $Q \subset X$ be countable. Given functions $R: Q \to \mathbb{R}^+$ and $\phi: \mathbb{R}^+ \to \mathbb{R}^+$, we consider the set $E(Q, R, \phi)$ of points $x \in X$ that ``hit'' the…

For a family $\mathcal{F}$ of sets and a disjoint pair $A,B$ we let $\mathcal{F}(A,\overline{B})=\{F\in \mathcal{F}: A\subseteq F, ~B\cap F=\emptyset\}$. The \textbf{$(p,q)$-d\"omd\"od\"om} of a family $\mathcal{F}\subseteq 2^{[n]}$ is…

The notion of $(k,m)$-agreeable society was introduced by Deborah Berg et al.: a family of convex subsets of $\R^d$ is called $(k,m)$-agreeable if any subfamily of size $m$ contains at least one non-empty $k$-fold intersection. In that…