Related papers: A Generalized Tur\'an Problem and its Applications

Given graphs $H$ and $F$, $\mathrm{ex}(n,H,F)$ denotes the largest number of copies of $H$ in $F$-free $n$-vertex graphs. Let $\chi(H)<\chi(F)=r+1$. We say that $H$ is $F$-Tur\'an-stable if the following holds. For any $\varepsilon>0$ there…

The $3$-uniform tight $\ell$-cycle minus one edge $C_{\ell}^{3-}$ is the $3$-graph on $\ell$ vertices consisting of $\ell-1$ consecutive triples in the cyclic order. We show that for every integer $\ell \ge 5$ satisfying $\ell\not\equiv…

In this paper we raise a variant of a classic problem in extremal graph theory, which is motivated by a design of fractional repetition codes, a model in distributed storage systems. For any feasible positive integers $d\geq 3$, $n \geq 3$,…

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…

We study the following question: how few edges can we delete from any $H$-free graph on $n$ vertices in order to make the resulting graph $k$-colorable? It turns out that various classical problems in extremal graph theory are special cases…

Zykov showed in 1949 that among graphs on $n$ vertices with clique number $\omega(G) \le \omega$, the Tur\'an graph $T_{\omega}(n)$ maximizes not only the number of edges but also the number of copies of $K_t$ for each size $t$. The problem…

Tur\'{a}n's theorem is a cornerstone of extremal graph theory. It asserts that for any integer $r \geq 2$ every graph on $n$ vertices with more than ${\tfrac{r-2}{2(r-1)}\cdot n^2}$ edges contains a clique of size $r$, i.e., $r$ mutually…

For a $k$-uniform hypergraph $F$ let $\textrm{ex}(n,F)$ be the maximum number of edges of a $k$-uniform $n$-vertex hypergraph $H$ which contains no copy of $F$. Determining or estimating $\textrm{ex}(n,F)$ is a classical and central problem…

In the $\ell$-Coloring Problem, we are given a graph on $n$ nodes, and tasked with determining if its vertices can be properly colored using $\ell$ colors. In this paper we study below-guarantee graph coloring, which tests whether an…

For graphs $H$ and $F$, let $\text{ex}(n,H,F)$ be the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices. The study of this function, which generalizes the well-known Tur\'{a}n number of graphs, was systematically…

In a recent breakthrough, Kelley and Meka (FOCS 2023) obtained a strong upper bound on the density of sets of integers without nontrivial three-term arithmetic progressions. In this work, we extend their result, establishing similar bounds…

Alon and Shikhelman initiated the systematic study of the following generalized Tur\'an problem: for fixed graphs $H$ and $F$ and an integer $n$, what is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph? An edge-colored…

Szemer\'edi's regularity lemma is a powerful tool in graph theory. It states that for every large enough graph, there exists a partition of the edge set with bounded size such that most induced subgraphs are quasirandom. When the graph is a…

Let $F$ be a graph which contains an edge whose deletion reduces its chromatic number. For such a graph $F,$ a classical result of Simonovits from 1966 shows that every graph on $n\ge n_0(F)$ vertices with more than…

A $k$-path is a hypergraph P_k = e_1,e_2,...,e_k such that |e_i \cap e_j| = 1 if |j - i| = 1 and e_i \cap e_j is empty otherwise. A k-cycle is a hypergraph C_k = e_1,e_2,.. ,e_k obtained from a (k-1)-path e_1,e_2,...,e_{k-1} by adding an…

Let $C_{2k_1, 2k_2, \ldots, 2k_t}$ denote the graph obtained by intersecting $t$ distinct even cycles $C_{2k_1}, C_{2k_2}, \ldots, C_{2k_t}$ at a unique vertex. In this paper, we determine the unique graphs with maximum adjacency spectral…

Let $f(k)$ be the maximum possible chromatic number of a graph whose edge set can be partitioned into at most $k$ complete bipartite graphs. Alon, Saks, and Seymour conjectured that $f(k)=k+1$ for all $k$. While the conjecture was verified…

The family $\Sigma_r$ consists of all $r$-graphs with three edges $D_1,D_2,D_3$ such that $|D_1\cap D_2|=r-1$ and $D_1 \triangle D_2 \subseteq D_3$. A generalized triangle, $\mathcal{T}_r \in \Sigma_r$ is an $r$-graph on…

The Erd\H{o}s-Rothschild problem from 1974 asks for the maximum number of $s$-edge colourings in an $n$-vertex graph which avoid a monochromatic copy of $K_k$, given positive integers $n,s,k$. In this paper, we systematically study the…

We consider the algorithmic problem of generating each subset of $[n]:=\{1,2,\ldots,n\}$ whose size is in some interval $[k,l]$, $0\leq k\leq l\leq n$, exactly once (cyclically) by repeatedly adding or removing a single element, or by…