Triangle-Intersecting Families of Graphs

A family of graphs F is said to be triangle-intersecting if for any two graphs G,H in F, the intersection of G and H contains a triangle. A conjecture of Simonovits and Sos from 1976 states that the largest triangle-intersecting families of graphs on a fixed set of n vertices are those obtained by fixing a specific triangle and taking all graphs containing it, resulting in a family of size (1/8) 2^{n choose 2}. We prove this conjecture and some generalizations (for example, we prove that the same is true of odd-cycle-intersecting families, and we obtain best possible bounds on the size of the family under different, not necessarily uniform, measures). We also obtain stability results, showing that almost-largest triangle-intersecting families have approximately the same structure.Comment: 43 page

Entropy of convolutions on the circle

Given ergodic p-invariant measures {\mu_i} on the 1-torus T=R/Z, we give a sharp condition on their entropies, guaranteeing that the entropy of the convolution \muon converges to \log p. We also prove a variant of this result for joinings of full entropy on \T^\N. In conjunction with a method of Host, this yields the following. Denote \sig_q(x) = qx\pmod{1}. Then for every p-invariant ergodic \mu with positive entropy, \frac{1}{N}\sum_{n=0}^{N-1}\sig_{c_n}\mu converges weak^* to Lebesgue measure as N \goesto \infty, under a certain mild combinatorial condition on {c_k}. (For instance, the condition is satisfied if p=10 and c_k=2^k+6^k or c_k=2^{2^k}.) This extends a result of Johnson and Rudolph, who considered the sequence c_k = q^k when p and q are multiplicatively independent. We also obtain the following corollary concerning Hausdorff dimension of sum sets: For any sequence {S_i} of p-invariant closed subsets of T, if \sum \dim_H(S_i) / |\log\dim_H(S_i)| = \infty, then \dim_H(S_1 + \cdots + S_n) \goesto 1.Comment: 34 pages, published versio

The distribution of maxima of approximately Gaussian random fields

Motivated by the problem of testing for the existence of a signal of known parametric structure and unknown ``location'' (as explained below) against a noisy background, we obtain for the maximum of a centered, smooth random field an approximation for the tail of the distribution. For the motivating class of problems this gives approximately the significance level of the maximum score test. The method is based on an application of a likelihood-ratio-identity followed by approximations of local fields. Numerical examples illustrate the accuracy of the approximations.Comment: Published in at http://dx.doi.org/10.1214/07-AOS511 the Annals of Statistics (http://www.imstat.org/aos/) by the Institute of Mathematical Statistics (http://www.imstat.org

Isotropy in Group Cohomology

The analogue of Lagrangians for symplectic forms over finite groups is studied, motivated by the fact that symplectic G-forms with a normal Lagrangian N<G are in one-to-one correspondence, up to inflation, with bijective 1-cocycle data on the quotients G/N. This yields a method to construct groups of central type from such quotients, known as Involutive Yang-Baxter groups. Another motivation for the search of normal Lagrangians comes from a non-commutative generalization of Heisenberg liftings which require normality. Although it is true that symplectic forms over finite nilpotent groups always admit Lagrangians, we exhibit an example where none of these subgroups is normal. However, we prove that symplectic forms over nilpotent groups always admit normal Lagrangians if all their p-Sylow subgroups are of order less than p^8