Let F be a family of connected bipartite graphs, each with at least three vertices. A proper vertex colouring of a graph G with no bichromatic subgraph in F is F-free. The F-free chromatic number χ(G,F) of a graph G is the minimum number of colours in an F-free colouring of G. For appropriate choices of F , several well-known types of colourings fit into this framework, including acyclic colour...

A sequence (ai) of integers is weak Sidon or well-spread if the sums ai + aj , for i < j , are all different. Let f (N) denote the maximum integer n for which there exists a weak Sidon sequence 0 a1< · · ·<an N . Using an idea of Lindström [An inequality for B2-sequences, J. Combin. Theory 6 (1969) 211–212], we offer an alternate proof that f (N)<N1/2 + O(N1/4), an inequality due to Ruzsa [Solv...

A subset E of an infinite discrete group G is called (i) an Rw-set if any bounded function on G supported by E is weakly almost periodic, (ii) a weak p-Sidon set (1 ~ p < 2) if on II (E) the IP -norm is bounded by a constant times the maximal C·-norm of I\G) , (iii) a T-set if xE n E and Ex n E are finite whenever x of e, and (iv) an FT-set if it is a finite union of T-sets. In this paper, we s...

In this survey paper I will discuss some old and new problems of a somewhat unconventional nature . Proofs will not be given, but I hope they will appear soon in several joint papers of V. T. Sós, Sárközy, and myself. Some of the older results can be found with detailed proofs in the excellent book Sequences of Halberstam and Roth . First of all an old question of Sidon posed to me more than 50...

A subset of an additive abelian group is h-Sidon set if every element in the h-fold sumset hA has a unique representation as sum h not necessarily distinct elements A. Let \(\mathbf {F}\) be field characteristic 0 with nontrivial absolute value, and let \(A = \{a_i :i \in \mathbf {N} \}\) \(B \{b_i subsets {F}\). \(\varepsilon \{ \varepsilon _i:i \}\), where _i > 0\) for all \(i {N}\). The B \)...