Combinatorica

For 0<γ≤1 and graphsG andH, we writeG→γH if any γ-proportion of the edges ofG span at least one copy ofH inG. As customary, we writeC k for a cycle of lengthk. We show that, for every fixed integerl≥1 and real η>0, there exists a real constantC=C(l, η), such that almost every random graphG n, p withp=p(n)≥Cn −1+1/2l satisfiesG n,p→1/2+η C 2l+1. In particular, for any fixedl≥1 and η>0, this result implies the existence of very sparse graphsG withG →1/2+η C 2l+1.

We introduce a class of infinite graphic matroids that contains all the motivating examples and satisfies an extension of Tutte’s excluded minors characterisation of finite graphic matroids.We prove that its members can be represented by certain ‘graph-like’ topological spaces previously considered by Thomassen and Vella.

For a graphG withn vertices and average valencyt, Turán’s theorem yields the inequalityα≧n/(t+1) whereα denotes the maximum size of an independent set inG. We improve this bound for graphs containing no large cliques.

We consider the problem of factoring polynomials overGF(p) for those prime numbersp for which all prime factors ofp− 1 are small. We show that if we have a primitivet-th root of unity for every primet dividingp− 1 then factoring polynomials overGF(p) can be done in deterministic polynomial time.

In [8], a deep and elegant analysis shows that double hashing is asymptotically equivalent to the ideal uniform hashing up to a load factor of about 0.319. In this paper we show how a randomization technique can be used to develop a surprisingly simple proof of the result that this equivalence holds for load factors arbitrarily close to 1.

By means of a partite construction we present a short proof of the Galvin Ramsey property of the class of all finite graphs and of its strengthening proved in [5]. We also establish a generalization of those results. Further we show that for every positive integerm there exists a graphH which is Ramsey forK m and does not contain two copies ofK m with more than two vertices in common.

Given a pointx in a convex figureM, letγ(x) denote the number of all affine diameters ofM passing throughx. It is shown that, for a convex figureM, the following conditions are equivalent.

We show that if a finite point set $$P\subseteq {\mathbb {R}}^2$$ has the fewest congruence classes of triangles possible, up to a constant M, then at least one of the following holds. This provides evidence for two conjectures of Erdős. We use the result of Petridis–Roche–Newton–Rudnev–Warren on the structure of the affine group combined with classical results from additive combinatorics.