Related papers: Complete subgraphs in multipartite graphs

In this paper, we consider the Johnson's graphs. We study the extremal properties of the Johnson's graphs. Namely, we investigate the number of edges in an arbitrary subgraph of this graph. Namely, in this article we prove analogs of…

A non-uniform hypergraph $H=(V,E)$ consists of a vertex set $V$ and an edge set $E\subseteq 2^V$; the edges in $E$ are not required to all have the same cardinality. The set of all cardinalities of edges in $H$ is denoted by $R(H)$, the set…

The Tur\'an number of a $k$-uniform hypergraph $H$, denoted by $e{x_k}\left({n;H} \right)$, is the maximum number of edges in any $k$-uniform hypergraph $F$ on $n$ vertices which does not contain $H$ as a subgraph. Let…

We construct dense, triangle-free, chromatic-critical graphs of chromatic number $k$ for all $k\geq 4$. For $k\geq 6$ our constructions have $> (\frac{1}{4} -\varepsilon)n^2$ edges, which is asymptotically best possible by Tur\'an's…

We suggest a new type of problem about distances in graphs and make several conjectures. As a first step towards proving them, we show that for sufficiently large values of n and k, a graph on n vertices that has no three vertices at…

We study the Tur\'{a}n problem for highly symmetric bipartite graphs arising from geometric shapes and periodic tilings commonly found in nature. 1. The prism $C_{2\ell}^{\square}:=C_{2\ell}\square K_{2}$ is the graph consisting of two…

We show that the set $\Pi^{(k)}$ of Tur\'an densities of $k$-uniform hypergraphs has infinitely many accumulation points in $[0,1)$ for every $k \geq 3$. This extends an earlier result of ours showing that $\Pi^{(k)}$ has at least one such…

We prove that every $n$-vertex graph with at least $\binom{n}{2} - (n - 4)$ edges has a fractional triangle decomposition, for $n \ge 7$. This is a key ingredient in our proof, given in a companion paper, that every $n$-vertex $2$-coloured…

In 1941, Turan conjectured that the edge density of any 3-graph without independent sets on 4 vertices (Turan (3,4)-graph) is >= 4/9(1-o(1)), and he gave the first example witnessing this bound. Brown (1983) and Kostochka (1982) found many…

In the 1980s, Erd\H{o}s and S\'os initiated the study of Tur\'an problems with a uniformity condition on the distribution of edges: the uniform Tur\'an density of a hypergraph $H$ is the infimum over all $d$ for which any sufficiently large…

We show that for every integer $k\geq3$, the set of Tur\'an densities of $k$-uniform hypergraphs has an accumulation point in $[0,1)$. In particular, $1/2$ is an accumulation point for the set of Tur\'an densities of $3$-uniform…

Goodman proved that the sum of the number of triangles in a graph on $n$ nodes and its complement is at least $n^3/24$; in other words, this sum is minimized, asymptotically, by a random graph with edge density $1/2$. Erd\H{o}s conjectured…

The celebrated Hajnal-Szemer\'edi theorem gives the precise minimum degree threshold that forces a graph to contain a perfect K_k-packing. Fischer's conjecture states that the analogous result holds for all multipartite graphs except for…

The Tur\'an number $\mathrm{ex}(n,H)$ of a graph $H$ is the maximum number of edges in an $n$-vertex graph which does not contain $H$ as a subgraph. The Tur\'{a}n number of regular polyhedrons was widely studied in a series of works due to…

Extremal problems concerning the number of independent sets or complete subgraphs in a graph have been well studied in recent years. Cutler and Radcliffe proved that among graphs with $n$ vertices and maximum degree at most $r$, where $n =…

This paper is motivated by the question of how global and dense restriction sets in results from extremal combinatorics can be replaced by less global and sparser ones. The result we consider here as an example is Turan's theorem, which…

An \emph{$(n,k,t)$-graph} is a graph on $n$ vertices in which every set of $k$ vertices contains a clique on $t$ vertices. Tur\'an's Theorem, rephrased in terms of graph complements, states that the unique minimum $(n,k,2)$-graph is an…

A cograph is a graph that contains no induced path $P_4$ on four vertices or equivalently a graph that can be constructed from vertices by sum and product operations. We study the bipartite Tur\'an problem restricted to cographs: for fixed…

An extremal graph for a given graph $H$ is a graph on $n$ vertices with maximum number of edges that does not contain $H$ as a subgraph. Let $s,t$ be integers and let $H_{s,t}$ be a graph consisting of $s$ triangles and $t$ cycles of odd…

The planar Tur\'{a}n number of a given graph $H$, denoted by $ex_{\mathcal{P}}(n,H)$, is the maximum number of edges over all planar graphs on $n$ vertices that do not contain a copy of $H$ as a subgraph. Let $H_k$ be a friendship graph,…