Showing papers in "Journal of The London Mathematical Society-second Series in 1982"

A Lower Bound for Heilbronn'S Problem

Abstract: We disprove Heilbronn's conjecture—that N points lying in the unit disc necessarily contain a triangle of area less than c/N. Introduction We think that the best account of the problem can be achieved by simply copying the corresponding paragraphs from Roth's paper [6]. \"Let Pv, P2,..., Pn (where n ^ 3) be a distribution of n points in a (closed) disc of unit area, such that the minimum of the areas of the triangles PiPjPk (taken over 1 < i < j < k ^ n) assumes its maximum possible value A = A(n). \"Heilbronn conjectured that A(n) c2(n/t)(\\ogt) . Remark 1. For any 3-graph one has the Turan-type estimate a > n/(3t), since a random (spanned) subgraph of size n/(2t) expectedly contains n/(24t) edges; delete all vertices in edges (see Spencer [8]). This is sharp up to constant multiple, for the Turan 3-graph (n/t disjoint cliques of size t) has t ~ t/2 and a = 2n/t. Thus the condition that G is uncrowded improves the bound on a by a factor (log t). This is again sharp, as is shown by random 3-graphs (choose nt triples at random and delete the few short cycles). Remark 2. Lemma 1 is an analogue of Lemma 1 in [1] or Theorem 2 in [2], which state that for a 2-graph with average valency t = 2e/n the Turan bound a ^ n(t+\\) can be improved to a > (i/\\00)(n/t)\\ogt if only the graph is trianglefree. The proof will also be analogous to the (complicated) one in [1], which uses random methods, rather than to the simple inductive one in [2], which was thoroughly rewritten and simplified by Joel Spencer. The reader is challenged to give a simple, non-probabilistic proof for Lemma 1. 2. The proof of the theorem If we drop N points to the unit disc at random, then (as will be seen shortly) we can select half of them with smallest triangle c/N (an alternative proof for Erdos' lower bound). We shall improve on this method by dropping N points and then selecting an appropriate subset of N points. Define the numbers t and n by the implicit equations t = n / 1 0 0 , N = c2(n/t)(\\ogt) 112 . Set A = (1/200) t/n; then n = -Nt(\\ogty' 2 and A = c3(log0/iV 2 = Cl(\\ogN)/N 2 . Let us drop n points to the unit disc at random, independently of each other, each with a uniform distribution. We define a 3-graph G on these n points (as vertices) by {a, b, c} € G if the points a, b, c form a triangle of area less than A. Now the probability that three random points form a triangle of area less than A is less than 2 2 — d{rn) = — Inrdr = 32TIA < t/n r J r o 16 JANOS KOML6S, JANOS PINTZ AND ENDRE SZEMERED1 (fix two points at a distance r, and then average over r). Hence the expected number of triangles of area less than A is less than nt/6. Thus the expected value of f is less than t/2. Hence, by Markov's inequality (see §4), with probability greater than 1/2, weget a 3-graph with t < t. Now we show that, with large probability, only o(n) short cycles occur in this 3-graph. All calculations will be based on the simple remark (already used above) that once two vertices have been chosen at a distance r, in order to get a triangle of area less than A, the third point has to belong to a strip of area less than 8A/r. The number of pairs of points at a distance less than d = n~° 6 is, with large probability, less than We discard these points. The number of 2-cycles is, with large probability, less than 2 f A c 4 n 4 —Inrdr < c5t \\ogn < n . The number of simple 3-cycles is, with large probability, less than

The asymptotic number of unlabelled regular graphs

Abstract: Over ten years ago Wright [4] proved a fundamental theorem in the theory of random graphs. He showed that if M = M(n) is such that almost no labelled graph of order n and size M has two isolated vertices or two vertices of degree n — 1, then the number of labelled graphs of order n and size M divided by the number of unlabelled graphs of order n and size M is asymptotic to n\\. The result is best possible, since if the above ratio is asymptotic to n\\ then almost no labelled graph of order n and size M has a non-trivial automorphism. The aim of this paper is to prove the analogue of Wright's theorem for regular graphs. Random regular graphs have not been studied for long. The main reason for this is that until recently there was no asymptotic formula for the number of labelled regular graphs: such a formula was found by Bender and Canfield [1]. Even more recently, in [2] a model was given for the set of labelled regular graphs which makes the study of random regular graphs fairly accessible. In particular, a result in [3] implies that, for r ^ 3, almost every labelled r-regular graph has only the identity as its automorphism. Let r ^ 3 be fixed and let n -* oo in such a way that rn = 2m is even. Denote by S£r = S£KtT the set of r-regular graphs with vertex set V = { l ,2 , . . . ,n} . Write °llr = °Unr for the set of unlabelled r-regular graphs of order n. Put Lr = \\££?r\\ and Ur = \\%\\. We know from [1] that ^ (r2_1)/4(2m)! Lr 2m\\ •

Inequalities for the Angular Derivative of an Analytic Function in the Unit Disk

Abstract: Let $ be a function, analytic in the unit disk, D, that maps the unit disk into itself ((z) ^ 2). In this paper, we present some inequalities for the angular derivative of cj). The more important of these concern the derivative of $ at its fixed points in the closed unit disk. Since (f) and 0' need not be continuous in D we need to clarify the terms \"fixed point\" and \"derivative of ^ at a fixed point\".

The complex of curves on non-orientable surfaces

Abstract: Let &$F{M) c 0>(.R + —0) denote the projectivized space of measured foliations on a compact surface M with negative Euler characteristic, as studied by Thurston [3], and let ^ ^ ( M ) denote the subspace consisting of those foliations in which each boundary component is a leaf (containing at least one singularity). If M is the sum of g tori, d disks and p projective planes, then 0>PQ{M) s S " and &&{M) is the join of &&0(M) to a (d 1 )-simplex. There is a subcomplex Xo in ^^0{M) whose (n — l)-simplices consist of foliations obtained by "enlarging" n disjoint simple, closed, connected curves Cl 5 . . . , Cn, none of which bounds a disk, or is boundary parallel, and no two of which bound an annulus. A subcomplex X of ^^(M) can be defined in much the same way, except that we allow proper arcs as well as simple closed curves. The complexes Xo and X have an interesting structure in their own right; since they are preserved under diffeomorphism, their structure gives geometric insight into the structure of automorphisms of M. Unfortunately, their structure is quite complex, since Xo and X are rarely locally finite. Floyd and Hatcher [2] have constructed all the low-dimensional examples (and one of dimension 5) in the cases where M is orientable. The purpose of the present paper is two-fold. First, we show that when M is not orientable, Xo has some surprising qualities not found when M is orientable. Secondly, we construct the complexes for those remaining cases in which the foliation space is of dimension one or two. The case of the 3-punctured RP is particularly interesting and is the subject of §3. In this case Xo is the complex obtained from the tetrahedron by repeated star subdivision of all the faces but no edges. In the resulting complex the vertices represent all the 1-sided curves, and an interior point of each edge all the 2-sided curves. I would like to thank David Chillingworth for his very helpful comments on the original manuscript.