Related papers: On generalized Tur\'an results in height two poset…

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The random Tur\'an number $\mathrm{ex}(G^r_{n,p},\mathcal{F})$ is the maximum number of edges in an $\mathcal{F}$-free subgraph of $G^r_{n,p}$, where $G^r_{n,p}$ is the…

Fix a $k$-chromatic graph $F$. In this paper we consider the question to determine for which graphs $H$ does the Tur\'an graph $T_{k-1}(n)$ have the maximum number of copies of $H$ among all $n$-vertex $F$-free graphs (for $n$ large…

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs, and let $H$ be an $r$-uniform hypergraph. Then $H$ is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$ as a subhypergraph. The Tur\'{a}n number of…

A poset is said to be (2+2)-free if it does not contain an induced subposet that is isomorphic to 2+2, the union of two disjoint 2-element chains. Two elements in a poset are indistinguishable if they have the same strict up-set and the…

Fix graphs $F$ and $H$ and let $ex(n,H,F)$ denote the maximum possible number of copies of the graph $H$ in an $n$-vertex $F$-free graph. The systematic study of this function was initiated by Alon and Shikhelman [{\it J. Comb. Theory, B}.…

We consider a problem proposed by Linial and Wilf to determine the structure of graphs that allows the maximum number of $q$-colorings among graphs with $n$ vertices and $m$ edges. Let $T_r(n)$ denote the Tur\'{a}n graph - the complete…

Let $A,B$ be disjoint sets of sizes $n$ and $m$. Let ${\mathcal Q}$ be a family of quadruples, having $2$ elements from $A$ and $2$ from $B$, such that any subset $S \subseteq A \cup B$ with $|S|=7$, $|S \cap A| \geq 2$ and $|S \cap B| \geq…

Let $B_n$ be the poset generated by the subsets of $[n]$ with the inclusion as relation and let $P$ be a finite poset. We want to embed $P$ into $B_n$ as many times as possible such that the subsets in different copies are incomparable. The…

We study maximum antichains in two posets related to quiver representations. Firstly, we consider the set of isomorphism classes of indecomposable representations ordered by inclusion. For various orientations of the Dynkin diagram of type…

We prove a generalised Ramsey--Tur\'an theorem for matchings, which (a) simultaneously generalises the Cockayne--Lorimer Theorem (Ramsey for matchings) and the Erd\H{o}s--Gallai Theorem (Tur\'an for matchings), and (b) is a generalised…

Given a set $X$, the power set $\mathbb{P}(X)$, and a finite poset $P$, a family $F\subset \mathbb{P}(X)$ is said to be induced-$P$-free if there is no injection $\phi: P\rightarrow \mathbb{F}$ such that $\phi(p)\subseteq\phi(q)$ if and…

Let $\mathcal{F}$ be a family of graphs. A graph is called $\mathcal{F}$-free if it does not contain any member of $\mathcal{F}$. Generalized Tur\'{a}n problems aim to maximize the number of copies of a graph $H$ in an $n$-vertex…

Motivated by the paper of Axenovich and Walzer [2], we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring on elements of…

In this paper, we study the following classical question of extremal set theory: what is the maximum size of a family of subsets of $[n]$ such that no $s$ sets from the family are pairwise disjoint? This problem was first posed by Erd\H os…

The problem of bounding the size of a set system under various intersection restrictions has a central place in extremal combinatorics. We investigate the maximum number of disjoint pairs a set system can have in this setting. In…

The crown $\Oh_{2t}$ is a height-2 poset whose Hasse diagram is a cycle of length $2t$. A family $\F$ of subsets of $[n]:=\{1,2..., n\}$ is {\em $\Oh_{2t}$-free} if $\Oh_{2t}$ is not a weak subposet of $(\F,\subseteq)$. Let…

We study the following family of problems: Given a set of $n$ points in convex position, what is the maximum number triangles one can create having these points as vertices while avoiding certain sets of forbidden configurations. As…

Let $ a,b \in {\bf Z}^+$, $r=a + b$, and let $T$ be a tree with parts $U = \{u_1,u_2,\dots,u_s\}$ and $V = \{v_1,v_2,\dots,v_t\}$. Let $U_1, \dots ,U_s$ and $V_1, \dots, V_t$ be disjoint sets, such that {$|U_i|=a$ and $|V_j|=b$ for all…

Given a graph G, let Q(G) denote the collection of all independent (edge-free) sets of vertices in G. We consider the problem of determining the size of a largest antichain in Q(G). When G is the edge-less graph, this problem is resolved by…

Let $P(n)$ be the set of all posets with $n$ elements. Let $P^{(j)}(n)$, $1\leq j\leq 2^n,$ be the number of all posets with $n$ elements possessing exactly $j$ antichains. We have determined the numbers $P^{(j)}(7),$ $1\leq j\leq 128$, and…