Related papers: The Zarankiewicz problem in 3-partite graphs

Since K\H{o}v\'ari, S\'os, and Tur\'an proved upper bounds for the Zarankiewicz problem in 1954, much work has been undertaken to improve these bounds, and some have done so by restricting to particular classes of graphs. In 2017, Fox,…

Let $C^{2k}_r$ be the $2k$-uniform hypergraph obtained by letting $P_1,...,P_r$ be pairwise disjoint sets of size $k$ and taking as edges all sets $P_i \cup P_j$ with $i \neq j$. This can be thought of as the `$k$-expansion' of the complete…

The expansion of a graph $F$, denoted by $F^3$, is the $3$-graph obtained from $F$ by adding a new vertex to each edge such that different edges receive different vertices. For large $n$, we establish tight upper bounds for: The maximum…

We expand Conlon's random algebraic construction to show that for any odd number $k \geq 3$ exists a natural number $c_k$ (the same as Conlon's) such that $\operatorname{ex}(n^a,n,\theta_{k,c_k}) = \Omega_{k,a}((n^{1 + a})^{\frac{k +…

Given a family of $k$-hypergraphs $\mathcal{F}$, $ex(n,\mathcal{F})$ is the maximum number of edges a $k$-hypergraph can have, knowing that said hypergraph has $n$ vertices but contains no copy of any hypergraph from $\mathcal{F}$ as a…

For graphs $H_1$ and $H_2$, if we glue them by identifying a given pair of vertices $u \in V(H_1)$ and $v \in V(H_2)$, what is the extremal number of the resulting graph $H_1^u \odot H_2^v$? In this paper, we study this problem and show…

In 2016, Dowden initiated the study of planar Tur\'an-type problems, which has since attracted considerable attention. Recently, Bekos et al. proved that every $K_3$-free $1$-planar graph on $n\ge 4$ vertices has at most $3n-6$ edges. In…

We prove that the number of multigraphs with vertex set $\{1, \ldots, n\}$ such that every four vertices span at most nine edges is $a^{n^2 + o(n^2)}$ where $a$ is transcendental (assuming Schanuel's conjecture from number theory). This is…

We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t, t}$ in the bipartite complement of $G$. Let $f(n, \Delta)$ be the largest $k$…

In this paper we present three different results dealing with the number of $(\leq k)$-facets of a set of points: 1. We give structural properties of sets in the plane that achieve the optimal lower bound $3\binom{k+2}{2}$ of $(\leq…

A graph is called an $(r,k)$-graph if its vertex set can be partitioned into $r$ parts, each having at most $k$ vertices and there is at least one edge between any two parts. Let $f(r,H)$ be the minimum $k$ for which there exists an…

A graph $G$ is $k$-critical (list $k$-critical, DP $k$-critical) if $\chi(G)= k$ ($\chi_\ell(G)= k$, $\chi_\mathrm{DP}(G)= k$) and for every proper subgraph $G'$ of $G$, $\chi(G')<k$ ($\chi_\ell(G')< k$, $\chi_\mathrm{DP}(G')<k$). Let $f(n,…

In 1975, Erd\H{o}s asked for the maximum number of edges that an $n$-vertex graph can have if it does not contain two edge-disjoint cycles on the same vertex set. It is known that Tur\'an-type results can be used to prove an upper bound of…

This paper gives tight upper bounds on the number of edges and the index for $\mathcal{K}^-_{r + 1}$-free unbalanced signed graphs, where $\mathcal{K}^-_{r + 1}$ is the set of $r+1$-vertices unbalanced signed complete graphs. \indent We…

There has been interest recently in maximizing the number of independent sets in graphs. For example, the Kahn-Zhao theorem gives an upper bound on the number of independent sets in a $d$-regular graph. Similarly, it is a corollary of the…

A central topic in extremal graph theory is the supersaturation problem, which studies the minimum number of copies of a fixed substructure that must appear in any graph with more edges than the corresponding Tur\'an number. Significant…

The codegree threshold $\mathrm{ex}_2(n, F)$ of a $3$-graph $F$ is the minimum $d=d(n)$ such that every $3$-graph on $n$ vertices in which every pair of vertices is contained in at least $d+1$ edges contains a copy of $F$ as a subgraph. We…

The $(k,r)$-fan is the graph consisting of $k$ copies of the complete graph $K_r$ which intersect in a single vertex, and is denoted by $F_{k,r}$. Erd\H{o}s, F\"uredi, Gould and Gunderson [J. Combin. Theory Ser. B 64 (1995) 89--100]…

For two $r$-graphs $\mathcal{T}$ and $\mathcal{H}$, let $\text{ex}_{r}(n,\mathcal{T},\mathcal{H})$ be the maximum number of copies of $\mathcal{T}$ in an $n$-vertex $\mathcal{H}$-free $r$-graph. The determination of Tur\'{a}n number…

For a given graph $F$, a graph $G$ is said to be $F$-saturated if $G$ contains no copy of $F$ but for any edge $uv\notin E(G)$, $G+uv$ contains a copy of $F$. The saturation number $sat(n,F)$ is defined as the minimum number of edges among…