New Computational Upper Bounds for Ramsey Numbers R(3,k)

Using computational techniques we derive six new upper bounds on the classical two-color Ramsey numbers: R(3,10) <= 42, R(3,11) <= 50, R(3,13) <= 68, R(3,14) <= 77, R(3,15) <= 87, and R(3,16) <= 98. All of them are improvements by one over the previously best known bounds. Let e(3,k,n) denote the minimum number of edges in any triangle-free graph on n vertices without independent sets of order k. The new upper bounds on R(3,k) are obtained by completing the computation of the exact values of e(3,k,n) for all n with k <= 9 and for all n <= 33 for k = 10, and by establishing new lower bounds on e(3,k,n) for most of the open cases for 10 <= k <= 15. The enumeration of all graphs witnessing the values of e(3,k,n) is completed for all cases with k <= 9. We prove that the known critical graph for R(3,9) on 35 vertices is unique up to isomorphism. For the case of R(3,10), first we establish that R(3,10) = 43 if and only if e(3,10,42) = 189, or equivalently, that if R(3,10) = 43 then every critical graph is regular of degree 9. Then, using computations, we disprove the existence of the latter, and thus show that R(3,10) <= 42.Comment: 28 pages (includes a lot of tables); added improved lower bound for R(3,11); added some note

Revisions

ABSTRACT: We present data which, to the best of our knowledge, includes all known nontrivial values and bounds for specific graph, hypergraph and multicolor Ramsey numbers, where the avoided graphs are complete or complete without one edge. Many results pertaining to other more studied cases are also presented. We give references to all cited bounds and values, as well as to previous similar compilations. We do not attempt complete coverage of asymptotic behavior of Ramsey numbers, but concentrate on their specific values

A case for a parallelizable hash

On November 2, 2007, NIST (United States National Institute of Standards and Technology) announced an initiative to design a new secure hash function for this century, to be called SHA-3. The competition will be open and it is planned to conclude in 2012. These developments are quite similar to the recent history of symmetric block ciphers— breaking of the DES (Data Encryption Standard) and emergence of the AES (Advanced Encryption Standard) in 2001 as the winner of a multiyear NIST competition. In this paper we make a case that parallelizability should be one of the properties sought in the new SHA-3 design. We present a design concept for a parallelizable hash function called PHASH based on a block cipher, and we discuss PHASH’s performance and security. 1