Bollobás, Erdős, and Szemerédi (1975) [1] investigated a tripartite generalization of the Zarankiewicz problem: what minimum degree forces graph with n vertices in each part to contain an octahedral K3(2)? They proved that n+2−1/2n3/4 suffices suggested it could be weakened n+cn1/2 for some constant c>0. In this note we show their method only gives n+(1+o(1))n11/12 provide many constructions if...

We show tight necessary and sufficient conditions on the sizes of small bipartite graphs whose union is a larger bipartite graph that has no large bipartite independent set. Our main result is a common generalization of two classical results in graph theory: the theorem of Kővári, Sós and Turán on the minimum number of edges in a bipartite graph that has no large independent set, and the theore...

For every positive integer k, we construct an explicit family of functions f : {0, 1}n → {0, 1} which has (k+ 1)-party communication complexity O(k) under every partition of the input bits into k + 1 parts of equal size, and k-party communication complexity Ω ( n k42k ) under every partition of the input bits into k parts. This improves an earlier hierarchy theorem due to V. Grolmusz. Our const...

Let F be a family of graphs. A graph is F-free if it contains no copy of a graph in F as a subgraph. A cornerstone of extremal graph theory is the study of the Turán number ex(n,F), the maximum number of edges in an F-free graph on n vertices. Define the Zarankiewicz number z(n,F) to be the maximum number of edges in an F-free bipartite graph on n vertices. Let Ck denote a cycle of length k, an...

Fix integers n ≥ r ≥ 2. A clique partition of ( [n] r ) is a collection of proper subsets A1, A2, . . . , At ⊂ [n] such that ⋃ i ( Ai r ) is a partition of ( [n] r ) . Let cp(n, r) denote the minimum size of a clique partition of ( [n] r ) . A classical theorem of de Bruijn and Erdős states that cp(n, 2) = n. In this paper we study cp(n, r), and show in general that for each fixed r ≥ 3, cp(n, ...

An order-s Davenport-Schinzel sequence over an n-letter alphabet is one avoiding immediate repetitions and alternating subsequences with length s+2. The main problem is to determine the maximum length of such a sequence, as a function of n and s. When s is fixed this problem has been settled (see Agarwal, Sharir, and Shor [1], Nivasch [12] and Pettie [15]) but when s is a function of n, very li...

Abstract Given a family $\mathcal{F}$ of bipartite graphs, the Zarankiewicz number $z(m,n,\mathcal{F})$ is maximum edges in an $m$ by $n$ graph $G$ that does not contain any member as subgraph (such called -free ). For $1\leq \beta \lt \alpha 2$ , graphs $(\alpha,\beta )$ - smooth if for some $\rho \gt 0$ and every $m\leq n$ $z(m,n,\mathcal{F})=\rho m n^{\alpha -1}+O(n^\beta . Motivated their w...