On restricted colourings of Kn

The authors investigate Ramsey-type extremal problems for finite graphs. In Section 1, anti-Ramsey numbers for paths are determined. For positive integers k and n let r=f(n,Pk) be the maximal integer such that there exists an edge colouring of Kn using precisely r colours but not containing any coloured path on k vertices with all edges having different colors. It is shown that f(n,P2k+3+ε)=t⋅n−(t+12)+1+ε for t≥5, n>c⋅t2 and ε=0,1. In Section 2, K3-spectra of colourings are determined. Given S⊆{1,2,3}, the authors investigate for which r and n there exist edge colourings of Kn using precisely r colours such that all triangles are s-coloured for some s∈S and, conversely, every s∈S occurs. Section 3 contains suggestions for further research

Intersection properties of subsets of integers

Let {A1,...,AN} be a family of subsets of {1, 2,...,n}. For a fixed integer k we assume that keyable*** is an arithmetic progression of ⩾k elements for every 1 ⩽ i < j ⩽ N. We would like to determine the maximum of N. For k = 0, R. L. Graham and the authors have proved that N=⩽(n3)+(n2)+(n1)+1For k ⩾ 2, the extremal and asymptotically extremal systems have (π224+o(1))n2setsFor k = 1, the maximum is between (n2)+1and(π224+12o(1))n2We conjecture that the lower bound is sharp

A hierarchy of randomness for graphs

AbstractIn this paper we formulate four families of problems with which we aim at distinguishing different levels of randomness.The first one is completely non-random, being the ordinary Ramsey–Turán problem and in the subsequent three problems we formulate some randomized variations of it. As we will show, these four levels form a hierarchy. In a continuation of this paper we shall prove some further theorems and discuss some further, related problems