Let G be a connected k–regular bipartite graph with bipartition V (G) = X ∪Y and adjacency matrix A. We say G is det–extremal if per(A) = |det(A)|. Det–extremal k–regular bipartite graphs exist only for k = 2 or 3. McCuaig has characterized the det–extremal 3–connected cubic bipartite graphs. We extend McCuaig’s result by determining the structure of det–extremal cubic bipartite graphs of conne...

In this paper we find, for n ≤ 16, the maximum number of edges in a 4-uniform hypergraph which does not have the complete 4-uniform hypergraph on five vertices,K4 5 , as a subgraph. Equivalently, we find all optimal (n, n−4, n−5) covering designs for n ≤ 16. Using these results we find a new upper bound for the Turán density ofK4 5 . π(K4 5 ) ≤ 1753 2380 = 0.73655 . . . . Finally wemake some no...

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

Extremal Combinatorics is an area in Discrete Mathematics that has developed spectacularly during the last decades. This paper contains a collection of problems and results in the area, including solutions or partial solutions to open problems suggested by various researchers in Extremal Graph Theory, Extremal Finite Set Theory and Combinatorial Geometry. This is not meant to be a comprehensive...

Recently, Souza introduced blowup Ramsey numbers as a generalization of bipartite numbers. For graphs $G$ and $H$, say $G\overset{r}{\longrightarrow} H$ if every $r$-edge-coloring contains monochromatic copy $H$. Let $H[t]$ denote the $t$-blowup Then number $G,H,r,$ $t$ is defined minimum $n$ such that $G[n] \overset{r}{\longrightarrow} H[t]$. proved upper lower bounds on are exponential in $t$...

Turán’s Theorem states that every graphG of edge density ‖G‖/ (|G| 2 ) > k−2 k−1 contains a complete graph K and describes the unique extremal graphs. We give a similar Theorem for `-partite graphs. For large `, we find the minimal edge density d` , such that every `-partite graph whose parts have pairwise edge density greater than d` contains a K . It turns out that d` = k−2 k−1 for large enou...

Given a 3-uniform hypergraph F , let ex3(n,F) denote the maximum possible size of a 3-uniform hypergraph of order n that does not contain any subhypergraph isomorphic to F . Our terminology follows that of [16] and [10], which are comprehensive survey articles of Turán-type extremal graph and hypergraph problems, respectively. Also see the monograph of Bollobás [2]. There is an extensive litera...