On Hypergraphs of Girth Five

In this paper, we study r-uniform hypergraphs H without cycles of length less than five, employing the definition of a hypergraph cycle due to Berge. In particular, for r = 3, we show that if H has n vertices and a maximum number of edges, then |H| = 1 6 n + o(n). This also asymptotically determines the generalized Turán number T3(n, 8, 4). Some results are based on our bounds for the maximum size of Sidon-type sets in Zn. 1 Definitions In this paper, a hypergraph H is a family of distinct subsets of a finite set. The members of H are called edges, and the elements of V (H) = ⋃ E∈HE are called vertices. If all edges in H have size r, then H is called an r-uniform hypergraph or, simply, r-graph. For example, a 2-graph is a graph in the usual sense. A vertex v and an edge E are called incident if v ∈ E. The degree of a vertex v of H, denoted d(v), is the number of edges ∗This work was initiated and continued at Microsoft Research during the author’s visits, and we are thankful the hosts for the opportunity and support. the electronic journal of combinatorics 10 (2003), #R25 1 of H incident with v. An r-graph H is r-partite if its vertex set V (H) can be colored in r colors in such a way that no edge of H contains two vertices of the same color. In such a coloring, the color classes of V (H) – the sets of all vertices of the same color – are called parts of H. We refer the reader to Berge [3] or [4] for additional background on hypergraphs. For k ≥ 2, a cycle in a hypergraph H is an alternating sequence of vertices and edges of the form v1, E1, v2, E2, . . . , vk, Ek, v1, such that (i) v1, v2, . . . , vk are distinct vertices of H (ii) E1, E2, . . . , Ek are distinct edges of H (iii) vi, vi+1 ∈ Ei for each i ∈ {1, 2, . . . , k − 1}, and vk, v1 ∈ Ek. We refer to a cycle with k edges as a k-cycle, and denote the family of all k-cycles by Ck. For example, a 2-cycle consists of a pair of vertices and a pair of edges such that the pair of vertices is a subset of each edge. The above definition of a hypergraph cycle is the “classical” definition (see, for example, Berge [3], [4], Duchet [11]). For r = 2 and k ≥ 3, it coincides with the definition of a cycle Ck in graphs and, in this case, Ck is a family consisting of precisely one member. Detailed discussions of alternative definitions of cycles in hypergraphs and the merits of each, as well as their applications in computer science, may be found in Duke [12] and Fagin [18]. The girth of a hypergraph H, containing a cycle, is the minimum length of a cycle in H. On a connection between 3-graphs of girth at least five and Greechie diagrams in quantum physics, see McKay, Megill and Pavičić [24]. 2 Problems and Results The topic of this paper falls into the context of Turán-type extremal problems in hypergraphs, on which an excellent survey was given by Füredi [19]. The question we consider is to determine the maximum number of edges in an r-graph on n vertices of girth five. For graphs (r = 2), this is an old problem of Erdős [14], which has its origins in a seminal paper of Erdős [13]. The best known lower and upper bounds are (1/2 √ 2)n+O(n) and (1/2)(n− 1)n, respectively. For bipartite graphs, on the other hand, this maximum is (1/2 √ 2)n+O(n) as n→∞. Many attempts at reducing the gap between the constants 1/2 √ 2 and 1/2 in the lower and upper bounds have not succeeded thus far (see Garnick, Kwong, Lazebnik [20] for more details). Surprisingly, we are able to obtain upper and lower bounds for the corresponding problem in 3-graphs which have equal leading terms. the electronic journal of combinatorics 10 (2003), #R25 2 Theorem 2.1 Let H be a 3-graph on n vertices and of girth at least five. Then |H| ≤ 1 6 n √ n− 3 4 + 1 12 n. For any odd prime power q ≥ 27, there exist 3-graphs H on n = q vertices, of girth five, with |H| = ( q+1 3 ) = 1 6 n − 1 6 n. This result is surprising in the sense that Turán-type questions for hypergraphs are generally harder than for graphs. One may formally apply the famous Ray-Chaudhuri and Wilson Theorem [25] to hypergraphs of girth at least three, which states that an r-graph, without a pair of sets intersecting in at least two points, has at most ( n 2 ) / ( r 2 ) edges, and the equality is attained for each r ≥ 3 and infinitely many n. Following de Caen [10], the generalized Turán number Tr(n, k, l) is defined to be the maximum number of edges in an r-graph on n vertices in which no set of k vertices spans l or more edges (or, equivalently, the union of any l edges contains more than k vertices). To illustrate this definition, the above-mentioned result of Ray-Chadhuri and Wilson is equivalent to the statement Tr(n, 2r − 2, 2) = ( n 2 ) / ( r 2 ) for each r ≥ 3 and infinitely many n. The problem of estimating Tr(n, k, l) in general was first approached by Brown, Erdős, and T. Sós [8], [9], who gave bounds for T3(n, k, l) for all k ≤ 6 and l ≤ 9, and established the asymptotics of the generalized Turán numbers T3(n, k, l) for (k, l) ∈ {(5, 3), (5, 4), (6, 4)}. In the case (k, l) = (6, 3), they established T3(n, 6, 3) > cn for some constant c. Remarkably precise bounds for T3(n, 6, 3) were given by Ruzsa and Szemerédi, who proved that for some constant c > 0 and all ε > 0, 2−c √ log n ≤ T3(n, 6, 3) < εn. The asymptotic behaviour of the numbers Tr(n, k, l), in general, remains unknown, and seems to be difficult to determine. For example, perhaps one of the most famous problems in extremal combinatorics is to prove or disprove Turán’s conjecture, that T3(n, 4, 4) ∼ 5 9 ( n 3 ) , n→∞. We now continue to relate the problem of estimating the size of hypergraphs of given girth with certain generalized Turán numbers. It is easy to see that T3(n, 4, 2) and T3(n, 6, 3) are precisely the maximum number of edges in a 3-graph of girth three and four respectively. Similarly, T3(n, 8, 4) is precisely the maximum number of edges in a 3-graph of girth five. This is seen by directly checking that any four triples on a set of eight vertices span a hypergraph containing a cycle of length at most four. Together with Theorem 2.1, and results about the density of primes (see Huxley [21]), this implies: Corollary 2.2 As n→∞, T3(n, 8, 4) ∼ 16n . the electronic journal of combinatorics 10 (2003), #R25 3 Generalizing to r-graphs, r ≥ 2, we are able to establish the following: Theorem 2.3 Let H be an r-graph, r ≥ 2, on n vertices and of girth at least five. Then |H| ≤ 1 r(r−1)n 3/2 + r−2 2r(r−1)n+O(n −1/2). Moreover, if H is r-partite, with n vertices in each part, then |H| ≤ 1 √ r−1 n 3/2 + 1 2 n+O(n). Finally, for each r ≥ 2, there exist r-partite r-graphs of girth at least five, with n ≥ 8r vertices in each part and 1 4 r−4r/3n4/3 edges. The proof of Theorem 2.3 for r = 2 gives the best known upper bounds for the maximum number of edges for girth five graphs and bipartite graphs, namely 1 2 n √ n− 1 and 1 2 n(1 + √ 4n− 3), respectively. The latter expression is an upper bound on the Zarankiewicz number – the maximum size of a bipartite graph with each part having n vertices and without cycles of length four (see, Kővári, T. Sós, Turán [22] and Reiman [26]). The lower bound in Theorem 2.3 is a probabilistic one. Attempts to establish explicit and better lower bounds led us to a generalization of the notion of a Sidon set in Zn, and to the question of its maximum cardinality. We remind the reader that a Sidon set in Zn (or in Z) is a set in which no two distinct pairs of elements have the same difference (or, equivalently, the same sum). The reader is referred to Babai and Sós [2] for more details on Sidon sets. Our generalization, roughly, will disallow equality between small integer multiples of such differences, and we present it next. Let k be a positive integer and let n be relatively prime to all elements of {1, 2, . . . , k}. Let a1, a2, a3, a4 be integers in {−k,−k+1, . . . , 0, . . . , k−1, k} such that a1+a2+a3+a4 = 0. Let S be the collection of sets S ⊂ {1, 2, 3, 4} such that ∑ i∈S ai = 0 and ai 6= 0 for i ∈ S. Now consider the following equation over Zn with respect to x = (x1, x2, x3, x4): a1x1 + a2x2 + a3x3 + a4x4 = 0. (1) A solution x of (1) is called trivial if there exists a partition of {1, 2, 3, 4} into sets S, T ∈ S such that xi = xj for all i, j ∈ S and all i, j ∈ T . This is analagous to the definition of a trivial solution (over the integers) to an equation of the form a1x1 + a2x2 + · · ·+ akxk = 0 by Ruzsa [27]. For example, consider the equation x1 + x2 − x3 − x4 = 0. Then S consists of the sets {1, 3}, {2, 4}, {1, 4}, {2, 3} and {1, 2, 3, 4}. Therefore the trivial solutions are those with x1 = x3, x2 = x4, or x1 = x4, x2 = x3, or x1 = x2 = x3 = x4. A set with only trivial solutions to x1+x2−x3−x4 = 0 is precisely a Sidon set. As the second example, consider the electronic journal of combinatorics 10 (2003), #R25 4 the equation 2x1−3x2+x4 = 0. Then S consists of the set {1, 2, 4}. The trivial solutions are therefore those for which x1 = x2 = x4. A k-fold Sidon set is a set A ⊂ Zn such that the equation (1) has only trivial solutions in A. For example, a 1-fold Sidon set is a Sidon set in the usual sense. For a 2-fold Sidon set A, each of the equations below admits only trivial solutions with x1, x2, x3, x4 ∈ A: x1 − x2 + x3 − x4 = 0, x1 + x2 − 2x3 = 0, x1 − x2 + 2x3 − 2x4 = 0 The definition of a k-fold Sidon set also extends to the set {1, 2, . . . , n} ⊂ Z, in which case the condition that n is relatively prime to all integers in {1, 2, . . . , k} may be dropped. How large can a k-fold Sidon set A in Zn be? Let us first present an elementary upper bound. To each pair {a, a′} of distinct elements of A, we can associate the set {i(a − a′) | i ∈ {1, 2, . . . , k}}. Note that each set has k elements and, for distinct pairs, the corresponding sets are disjoint. It follows immediately that k (|A| 2 ) ≤ n and therefore |A| < (2n/k) + 1. To improve this bound we will use Theorem 2.3 in a way described below. Let A be a subset of Zn, and let B be a set of r integers. Define H(A,B) to be the r-partite r-graph with parts Vb = Zn, b ∈ B. For each x ∈ Zn and each a ∈ A, an edge of H(A,B) is the set of r vertices {x + ba : b ∈ B}, where x + ba ∈ Vb. Hence H(A,B) contains rn vertices and |A|n edges. The following proposition establishes a connection between r-partite r-graphs of girth five and k-fold Sidon sets. Proposition 2.4 Let n, k, r be positive integers, and n be odd. Let B ⊂ Z be a Sidon set of integers of size r such that all differences of distinct elements of B are relatively prime to n and do not exceed k. Let A be a k-fold Sidon set in Zn. Then H(A,B) is an r-partite r-graph of girth at least five, with |A|n edges. Theorem 2.3 and Proposition 2.4 sometimes lead to a better constant in the upper bound for the size of a k-fold Sidon set of Zn. For example, let k = 3, gcd(n, 6) = 1, and B = {−1, 0, 2} (a Sidon set). Then, applying Theorem 2.3 (with r = 3) and Proposition 2.4, we can reduce the bound (2n/3) on a 3-fold Sidon set to (n/2). Next, for infinitely many n, we provide a lower bound within 2 factor of the upper bound on the size of a 2-fold Sidon set: Theorem 2.5 Let t be a positive integer, and let n = 2 t+1 + 2 t + 1. Then, there exists a 2-fold Sidon set A in Zn, such that |A| ≥ 1 2 n − 3. the electronic journal of combinatorics 10 (2003), #R25 5 It seems likely that for each integer k ≥ 3, there exists a k-fold Sidon set in Zn (or in {1, 2, . . . , n} ⊂ Z) of size cn for some c > 0 depending only on k. By Theorem 2.5 and Theorem 2.3, we immediately obtain the following result: Theorem 2.6 Let H be a 3-partite 3-graph with n ≥ 3 vertices in each of its parts and of girth at least five. Then |H| ≤ 1 √ 2 n + n. Let i be a positive integer and let n = 2 i+1 + 2 i + 1. Then there exists a 3-partite 3-graph H, with n vertices in each part, of girth at least five, such that |H| ≥ 1 2 n − 3n. We remark that from the second part of Theorem 2.1, we obtain a weaker lower bound of (1+o(1)) √ 3 9 n, by applying the Erdős-Kleitmann Lemma [15] in the case r = 3: every r-graph H on rn vertices contains an r-partite r-graph with n vertices in each part and at least r! rr |H| edges.