For an ordered set W = {w1, w2, . . . , wk} of vertices in a connected graph G and a vertex v of G, the code of v with respect to W is the k-vector cW (v) = (d(v, w1), d(v, w2), . . . , d(v, wk)). The setW is an independent resolving set for G if (1)W is independent in G and (2) distinct vertices have distinct codes with respect to W . The cardinality of a minimum independent resolving set in G...

We derive bounds on the size of an independent set based on eigenvalues. This generalizes a result due to Delsarte and Hoffman. We use this to obtain new bounds on the independence number of the Erdős-Rényi graphs. We investigate further properties of our bounds, and show how our results on the Erdős-Rényi graphs can be extended to other polarity graphs.

imply the solvability of the equation a s-av = b' ; the notion of sum intersector set is defined similarly. The authors prove two general theorems saying that if a set {b, , b 2 BLOCKINbi} is well distributed simultaneously among and within all residue classes of small moduli then it must be both difference and sum intersector set. They apply these theorems to investigate the solvability of the...

The well-known Ramsey number r(t, u) is the smallest integer n such that every Ktfree graph of order n contains an independent set of size u. In other words, it contains a subset of u vertices with no K2. Erdős and Rogers introduced a more general problem replacing K2 by Ks for 2 ≤ s < t. Extending the problem of determining Ramsey numbers they defined the numbers fs,t(n) = min { max{|W | : W ⊆...

Let G be a graph on n vertices in which every induced subgraph on s = log n vertices has an independent set of size at least t = log n. What is the largest q = q(n) so that every such G must contain an independent set of size at least q ? This is one of several related questions raised by Erdős and Hajnal. We show that q(n) = Θ(log n/ log log n), investigate the more general problem obtained by...

Given a projective plane Σ and a polarity θ of Σ, the corresponding polarity graph is the graph whose vertices are the points of Σ, and two distinct points p1 and p2 are adjacent if p1 is incident to p θ 2 in Σ. A well-known example of a polarity graph is the Erdős-Rényi orthogonal polarity graph ERq, which appears frequently in a variety of extremal problems. Eigenvalue methods provide an uppe...

Let S ⊂ Rk+m be a compact semi-algebraic set defined by P1 ≥ 0, . . . , P` ≥ 0, where Pi ∈ R[X1, . . . , Xk, Y1, . . . , Ym], and deg(Pi) ≤ 2, 1 ≤ i ≤ `. Let π denote the standard projection from Rk+m onto Rm. We prove that for any q > 0, the sum of the first q Betti numbers of π(S) is bounded by (k +m). We also present an algorithm for computing the the first q Betti numbers of π(S), whose com...

A subgraph of a graph G is called trivial if it is either a clique or an independent set. Let q(G) denote the maximum number of vertices in a trivial subgraph of G. Motivated by an open problem of Erdős and McKay we show that every graph G on n vertices for which q(G) ≤ C log n contains an induced subgraph with exactly y edges, for every y between 0 and n. Our methods enable us also to show tha...

The zero-error capacity of a classical channel is expressed in terms of the independence number of some graph and its tensor powers. This quantity is hard to compute even for small graphs such as the cycle of length seven, so upper bounds such as the Lovász theta function play an important role in zero-error communication. In this paper, we show that the Lovász theta function is an upper bound ...