John Talbot

Hypergraphs do jump

We say that α ∈ [0, 1) is a jump for an integer r ≥ 2 if there exists c(α) > 0 such that for all ϵ > 0 and all t ≥ 1, any r-graph with n ≥ n0(α, ϵ, t) vertices and density at least α + ϵ contains a subgraph on t vertices of density at least α + c. n nThe Erdős–Stone–Simonovits theorem [4, 5] implies that for r = 2, every α ∈ [0, 1) is a jump. Erdős [3] showed that for all r ≥ 3, every α ∈ [0, r!/rr) is a jump. Moreover he made his famous ‘jumping constant conjecture’, that for all r ≥ 3, every α ∈ [0, 1) is a jump. Frankl and Rodl [7] disproved this conjecture by giving a sequence of values of non-jumps for all r ≥ 3. n nWe use Razborovs flag algebra method [9] to show that jumps exist for r = 3 in the interval [2/9, 1). These are the first examples of jumps for any r ≥ 3 in the interval [r!/rr, 1). To be precise, we show that for r = 3 every α ∈ [0.2299, 0.2316) is a jump. n nWe also give an improved upper bound for the Turan density of K4− = {123, 124, 134}: π(K4−) ≤ 0.2871. This in turn implies that for r = 3 every α ∈ [0.2871, 8/27) is a jump.

Complexity and Cryptography: An Introduction

Cryptography plays a crucial role in many aspects of todays world, from internet banking and ecommerce to email and web-based business processes. Understanding the principles on which it is based is an important topic that requires a knowledge of both computational complexity and a range of topics in pure mathematics. This book provides that knowledge, combining an informal style with rigorous proofs of the key results to give an accessible introduction. It comes with plenty of examples and exercises (many with hints and solutions), and is based on a highly successful course developed and taught over many years to undergraduate and graduate students in mathematics and computer science.

Intersecting Families of Separated Sets

A set

Compression and Erdős-Ko-Rado graphs

Asubseteq {1,2,ldots,n}

Graphs with the Erdős-Ko-Rado property

is said to be

A note on the jumping constant conjecture of Erdős

k

A solution to the 2/3 conjecture

- separated if, when considered on the circle, any two elements of

Chromatic Turán problems and a new upper bound for the Turán density of K4

A

Vertex Turán problems in the hypercube

are separated by a gap of size at least

The minimal density of triangles in tripartite graphs

k