Projections of Bodies and Hereditary Properties of Hypergraphs

We prove that for every M-dimensional body K, there is a rectangular parallelepiped B of the same volume as K, such that the projection of B onto any coordinate subspace is at most as large as that of the corresponding projection of K. We apply this theorem to projections of finite set systems and to hereditary properties. In particular, we show that every hereditary property of uniform hypergraphs has a limiting density. 1. Projections of bodies Let AT be a body in U, and let (u19., vn) be the standard basis for IR . Denote the volume of K by \K\. Furthermore, given a subset A e [n] — {1,2,. ,«} with d elements, denote by KA the orthogonal projection of K onto the subspace spanned by {vt'. ieA}, and by \KA\ its (//-dimensional) volume. Thus KM = K. By the term box we shall mean a rectangular parallelepiped whose sides are parallel to the coordinate axes. For the purposes of this paper, a body can be taken to be a compact subset of U which is the closure of its interior. It would be effortless to rewrite our results and their proofs in terms of arbitrary product measures and measurable subsets of U, using outer measures on the projections. We choose not to write down these extensions, in order to avoid irrelevant technicalities cluttering up an otherwise simply stated theorem, which is the following. THEOREM 1. Let K be a body in U. Then there is a box B in U with \B\ = \K\ and \BA\^\KA\foreveryA^[n]. It is immediate that if bounds on the volumes of the (n — l)-dimensional projections of a box B are given, then a bound on the total volume ensues, because I~l"-i l̂ cmxwl = l^l""From Theorem 1, it then follows that the inequality n<-i l̂ [n]\«l ^ I-Kp" holds for any body K. This is the Loomis-Whitney inequality [14] (see also [7, page 95] and [12, page 162]; the inequality was rediscovered by Allan [3], who gave a more streamlined proof). A valuable consequence of Theorem 1 is that if the volume of a box can be bounded in terms of the volumes of a certain collection of projections, then the same bound will be valid for all bodies. We shall, in fact, prove Theorem 1 by examining some collections of projections for which there is such a bound on the volume of a box. By a cover of [n] we mean a multiset # of subsets of [n] such that each element ie\n] is in at least one of the members of €. A k-cover is a cover in which each element Received 22 June 1993; revised 26 September 1993, 15 January 1994. 1991 Mathematics Subject Classification 05D05, 05C30, 05C65, 05C80, 28A75, 51M16, 51M25. Bull. London Math. Soc. 27 (1995) 417-424 418 BELA BOLLOBAS AND ANDREW THOMASON of [n] is in exactly k of the members of <%. We emphasise that the sets in a cover need not be distinct; for example, {[«],[«],[«]} is a 3-cover of [«]. A uniform cover of [n] is a A>cover for some k ^ 1. The 1-cover {[«]} of [n] is said to be trivial, all other covers being non-trivial. A uniform cover of [n] which is not the disjoint union of two uniform covers of [n] is said to be irreducible. It is important to note that there are only a finite number of irreducible uniform covers of [«]. One way of seeing this is to consider an infinite sequence Fo = C^j/O^i of distinct uniform covers of [n]. Enumerate the subsets AX,A2, . , A2n of [n], and then select infinite sequences F( = Ctfj'O^u 1 ^ l ^ 2, so that Ft is a subsequence of Ti_1 and so that the number of copies of At in ^\ j) is a nondecreasing function of/ It is easy to see this can be done, and it is clear from the construction that the covers in F2n are nested. Hence if # ' and # are any two terms of F2n, with # ' ^ €, then # is not irreducible, it being the disjoint union of the two uniform covers W and # \ # ' . In fact, writing D{n) for the number of irreducible covers of [n], Huckeman, Jurkat and Shapley proved (see Graver [11]) that D(n) ^ (n + l) for all n. Related results have been proved by Alon and Berman [4] and Fiiredi [10]. Clearly, if B is a box and <# is a fc-cover of [n], then JXietfl^J = l-̂ l*I n ^ w °f this, the next result, Theorem 2, can be considered as a special case of Theorem 1. However, in order to prove Theorem 1, we must first prove Theorem 2 directly and use it to derive Theorem 1. THEOREM 2. Let K be a body in U, and let <€ be a k-cover of[n]. Then i f i t **• i r i f i |A| ^ 1 1 \KA\. Proof. The proof is by induction on n, the case n = 1 being trivial. For the general case, for each xe U let K(x) be the section of K consisting of points with nth coordinate equal to x, so that 1*1 = j\K(x)\dx. We define W = {AeW: neA} and T = <i\i\ so that |tf'| = k. Then {A\{n}:AeT}[) <€" forms a A>cover of [n-1], so, by the induction hypothesis,