On large FF-Diophantine sets

Let $F\in\mathbb{Z}[x,y]$ and $m\ge2$ be an integer. A set $A\subset \mathbb{Z}$ is called an $(F,m)$-Diophantine set if $F(a,b)$ is a perfect $m$-power for any $a,b\in A$ where $a\ne b$. If $F$ is a bivariate polynomial for which there exist infinite $(F,m)$-Diophantine sets, then there is a complete qualitative characterization of all such polynomials $F$. Otherwise, various finiteness results are known. We prove that given a finite set of distinct integers $S$ of size $n$, there are infinitely many bivariate polynomials $F$ such that $S$ is an $(F,2)$-Diophantine set. In addition, we show that the degree of $F$ can be as small as $\displaystyle 4\lfloor n/3\rfloor$

Borel sets with large squares

For a cardinal mu we give a sufficient condition (*)_mu (involving ranks measuring existence of independent sets) for: [(**)_mu] if a Borel set B subseteq R x R contains a mu-square (i.e. a set of the form A x A, |A|= mu) then it contains a 2^{aleph_0}-square and even a perfect square, and also for [(***)_mu] if psi in L_{omega_1, omega} has a model of cardinality mu then it has a model of cardinality continuum generated in a nice, absolute way. Assuming MA + 2^{aleph_0}> mu for transparency, those three conditions ((*)_mu, (**)_mu and (***)_mu) are equivalent, and by this we get e.g.: for all alpha= aleph_alpha => not (**)_{aleph_alpha}, and also min {mu :(*)_mu}, if <2^{aleph_0}, has cofinality aleph_1. We deal also with Borel rectangles and related model theoretic problems

Large isoperimetric surfaces in initial data sets

We study the isoperimetric structure of asymptotically flat Riemannian 3-manifolds (M,g) that are C^0-asymptotic to Schwarzschild of mass m>0. Refining an argument due to H. Bray we obtain an effective volume comparison theorem in Schwarzschild. We use it to show that isoperimetric regions exist in (M, g) for all sufficiently large volumes, and that they are close to centered coordinate spheres. This implies that the volume-preserving stable constant mean curvature spheres constructed by G. Huisken and S.-T. Yau as well as R. Ye as perturbations of large centered coordinate spheres minimize area among all competing surfaces that enclose the same volume. This confirms a conjecture of H. Bray. Our results are consistent with the uniqueness results for volume-preserving stable constant mean curvature surfaces in initial data sets obtained by G. Huisken and S.-T. Yau and strengthened by J. Qing and G. Tian. The additional hypotheses that the surfaces be spherical and far out in the asymptotic region in their results are not necessary in our work.Comment: 29 pages. All comments welcome! This is the final version to appear in J. Differential Geo

Probabilistic Existence of Large Sets of Designs

A new probabilistic technique for establishing the existence of certain regular combinatorial structures has been recentlyintroduced by Kuperberg, Lovett, and Peled (STOC 2012). Using this technique, it can be shown that under certain conditions, a randomly chosen structure has the required properties of a $t$-$(n,k,\lambda)$ combinatorial design with tiny, yet positive, probability. The proof method of KLP is adapted to show the existence of large sets of designs and similar combinatorial structures as follows. We modify the random choice and the analysis to show that, under the same conditions, not only does a $t$-$(n,k,\lambda)$ design exist but, in fact, with positive probability there exists a large set of such designs -- that is, a partition of the set of $k$-subsets of $[n]$ into $t$-designs $t$-$(n,k,\lambda)$ designs. Specifically, using the probabilistic approach derived herein, we prove that for all sufficiently large $n$, large sets of $t$-$(n,k,\lambda)$ designs exist whenever $k > 12t$ and the necessary divisibility conditions are satisfied. This resolves the existence conjecture for large sets of designs for all $k > 12t$.Comment: 20 page