Vera T. Sós

Sidon Sets in Groups and Induced Subgraphs of Cayley Graphs

Let S be a subset of a group G. We call S a Sidon subset of the first (second) kind, if for any x, y, z, w ∈ S of which at least 3 are different, xy ≠ zw (xy-1 ≠ zw-1, resp.). (For abelian groups, the two notions coincide.) If G has a Sidon subset of the second kind with n elements then every n-vertex graph is an induced subgraph of some Cayley graph of G. We prove that a sufficient condition for G to have a Sidon subset of order n (of either kind) is that (❘G❘ ⩾ cn3. For elementary Abelian groups of square order, ❘G❘ ⩾ n2 is sufficient. We prove that most graphs on n vertices are not induced subgraphs of any vertex transitive graph with We warn the reader that the sets considered in this paper are different from the Sidon sets Fourier analysts investigate.

Maximum induced trees in graphs

G that induces a tree. We investigate the relationship of t(G)to other parameters associated with G : the number of vertices and edges, the radius, the independence number, maximum clique size and connectivity . The central result is a set of upper and lower bounds for the function f(n, p), defined to be the minimum ofover all connected graphs with n vertices andn-1 +p edges. The bounds obtained yield an asymptotic characterization of the function correct to leading order in almost all ranges. The results show that f(n, p)is surprisingly small ; in particular ,fin, (n)=2loglogn+OfogloglogP) for any constant c>0, and fin,tiT ) = 2log(I + 1 ~) ,)± 4 for 0< 1 and n sufficiently large . Bounds ont(G)are obtained in terms of the size of the largest clique . These are used to formulate bounds for a Ramsey-type function, N(k, t), the smallest integer so that every connected graph on N(k, t) vertices has either a clique of size k or an induced tree of size t.Tight bounds fort(G)from the independence number a(G)are also proved . It is shown that every connected graph with radius r has an induced path, and hence an induced tree, on 2r- I vertices .

On restricted colourings of K n

AbstractGiven a sample graphH and two integers,n andr, we colourKn byr colours and are interested in the following problem.n Which colourings of the subgraphs isomorphic to H in Knmust always occur (and which types of colourings can occur whenKn is coloured in an appropriate way)?These types of problems include theRamsey theory, where we ask: for whichn andr must a monochromaticH occur. They also include theanti-Ramsey type problems, where we are trying to ensure a totally multicoloured copy ofH, that is, anH each edge of which has different colour.

On Ramsey—Turán type theorems for hypergraphs

AbstractLetHr be anr-uniform hypergraph. Letg=g(n;Hr) be the minimal integer so that anyr-uniform hypergraph onn vertices and more thang edges contains a subgraph isomorphic toHr. Lete =f(n;Hr,εn) denote the minimal integer such that everyr-uniform hypergraph onn vertices with more thane edges and with no independent set ofεn vertices contains a subgraph isomorphic toHr.We show that ifr>2 andHr is e.g. a complete graph thenn

Intersection Theorems on Structures

mathop {lim }limits_{varepsilon to 0} mathop {lim }limits_{n to infty } left( {begin{array}{*{20}c} n r end{array} } right)^{ - 1} f(n;H^r ,varepsilon n) = mathop {lim }limits_{n to infty } left( {begin{array}{*{20}c} n r end{array} } right)^{ - 1} g(n;H^r )

Largest digraphs contained in alln-tournaments

n while for someHr withn

Intersection theorems for t -valued functions

mathop {lim }limits_{n to infty } left( {begin{array}{*{20}c} n r end{array} } right)^{ - 1} g(n;H^r ) ne 0