An extremal theorem in the hypercube

The hypercube Q_n is the graph whose vertex set is {0,1}^n and where two vertices are adjacent if they differ in exactly one coordinate. For any subgraph H of the cube, let ex(Q_n, H) be the maximum number of edges in a subgraph of Q_n which does not contain a copy of H. We find a wide class of subgraphs H, including all previously known examples, for which ex(Q_n, H) = o(e(Q_n)). In particular, our method gives a unified approach to proving that ex(Q_n, C_{2t}) = o(e(Q_n)) for all t >= 4 other than 5.Comment: 6 page

Extremal numbers for cycles in a hypercube

Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} + \frac{1}{3(2k-2)}} 2^n).$$Comment: New reference [18] for a better bound by I.Tomon is adde

Upper bounds on the size of 4- and 6-cycle-free subgraphs of the hypercube

In this paper we modify slightly Razborov's flag algebra machinery to be suitable for the hypercube. We use this modified method to show that the maximum number of edges of a 4-cycle-free subgraph of the n-dimensional hypercube is at most 0.6068 times the number of its edges. We also improve the upper bound on the number of edges for 6-cycle-free subgraphs of the n-dimensional hypercube from the square root of 2 - 1 to 0.3755 times the number of its edges. Additionally, we show that if the n-dimensional hypercube is considered as a poset, then the maximum vertex density of three middle layers in an induced subgraph without 4-cycles is at most 2.15121 times n choose n/2.Comment: 14 pages, 9 figure

A class of graphs of zero Tur\'an density in a hypercube

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is at least a positive proportion of the number of edges in $Q_n$, $H$ is said to have a positive Tur\'an density in a hypercube or simply a positive Tur\'an density; otherwise it has a zero Tur\'an density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has a positive or a zero Tur\'an density remains a widely open question for general $H$. By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of cubical graphs, ones having so-called partite representation, that have a zero Tur\'an density. He raised a question whether this gives a characterisation, i.e., whether a cubical graph has zero Tur\'an density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of cubical graphs which have no partite representation, but on the other hand, have a zero Tur\'an density. In addition, we show that any graph whose every block has partite representation has a zero Tur\'an density in a hypercube