For every positive integer n, we construct a Hasse diagram with n vertices and independence number O ( 3 / 4 ) . Such graphs have chromatic Ω 1 , which significantly improves the previously best-known constructions of diagrams having Θ log In addition, if also require girth at least k ⩾ 5 such most − 2 + o The proofs are based on existence point-line arrangements in plane many incidences avoids...

One of the most elementary consequences of the assumption that a normal division algebra A is cyclic of degree n over its centrum K is that A contains a quantity j whose minimum equation is co=7 in K. In 1933 I conjectured the truth of the converse proposition. The proof is easily reducible f to the case where n is a power p of a prime p. Let a be the characteristic of K. I succeeded in proving...

The class Av(1324), of permutations avoiding the pattern 1324, is one of the simplest sets of combinatorial objects to define that has, thus far, failed to reveal its enumerative secrets. By considering certain large subsets of the class, which consist of permutations with a particularly regular structure, we prove that the growth rate of the class exceeds 9.81. This improves on a previous lowe...

In 1947 D. H. Lehmer conjectured that Ramanujan’s tau-function never vanishes. In the 1980s, B. Gross and D. Zagier proved a deep formula expressing the central derivative of suitable Hasse-Weil L-functions in terms of the Neron-Tate height of a Heegner point. This expository article describes recent work (with J. H. Bruinier and R. Rhoades) which reformulates both topics in terms of the algebr...

For any number field we calculate the exact proportion of rational numbers which are everywhere locally a norm but not globally a norm from the number field.