Related papers: Counting substructures and eigenvalues I: triangle…

Given a graph $G$, let $\tau_1(G)$ denote the smallest size of a set of edges whose deletion makes $G$ triangle-free, and let $\alpha_1(G)$ denote the largest size of an edge set containing at most one edge from each triangle of $G$.…

A connected $k$-chromatic graph $G$ with $k \geq 3$ is said to be triangle-critical, if every edge of $G$ is contained in an induced triangle of $G$ and the removal of any triangle from $G$ decreases the chromatic number of $G$ by three. B.…

A celebrated but non-effective theorem of Tibor Gallai states that for any finite set $A$ of $\Z^n$ and for any finite number of colors $c$ there is a minimal $m$ such that no coloring of the finite $m^n$-grid can avoid that a homothetic…

For a sequence $(H_i)_{i=1}^k$ of graphs, let $\textrm{nim}(n;H_1,\ldots, H_k)$ denote the maximum number of edges not contained in any monochromatic copy of $H_i$ in colour $i$, for any colour $i$, over all $k$-edge-colourings of~$K_n$.…

In this paper, we first study a new extremal problem recently posed by Conlon and Tyomkyn~(arXiv: 2002.00921). Given a graph $H$ and an integer $k\geqslant 2$, let $f_{k}(n,H)$ be the smallest number of colors $c$ such that there exists a…

An $(a:b)$-coloring of a graph $G$ is a function $f$ which maps the vertices of $G$ into $b$-element subsets of some set of size $a$ in such a way that $f(u)$ is disjoint from $f(v)$ for every two adjacent vertices $u$ and $v$ in $G$. The…

The classical Andr\'{a}sfai-Erd\H{o}s-S\'{o}s theorem considers the chromatic number of $K_{r + 1}$-free graphs with large minimum degree, and in the case $r = 2$ says that any $n$-vertex triangle-free graph with minimum degree greater than…

The Ruzsa-Szemer\'{e}di $(6,3)$-problem can be equivalently stated as determining the maximum number of edge-disjoint triangles on $n$ vertices such that no triangle is formed by edges from three distinct triangle-copies. Gowers and Janzer…

Suppose that $R$ (red) and $B$ (blue) are two graphs on the same vertex set of size $n$, and $H$ is some graph with a red-blue coloring of its edges. How large can $R$ and $B$ be if $R\cup B$ does not contain a copy of $H$? Call the largest…

A simple probabilistic argument shows that every $r$-uniform hypergraph with $m$ edges contains an $r$-partite subhypergraph with at least $\frac{r!}{r^r}m$ edges. The celebrated result of Edwards states that in the case of graphs, that is…

Let $G = (V,E)$ be a simple graph and let $\{R,B\}$ be a partition of $E$. We prove that whenever $|E| + \min\{ |R|, |B| \} > { |V| \choose 2 }$, there exists a subgraph of $G$ isomorphic to $K_3$ which contains edges from both $R$ and $B$.…

Keevash, Lenz and Mubayi developed a general criterion for hypergraph spectral extremal problems in their seminal work (SIAM J. Discrete Math., 2014). Their framework shows that extremal results on the $\alpha$-spectral radius (for $\alpha…

A graph is called $F$-free if it does not contain a copy of $F$. Let $G(r,s)$ denote a $K_{r+1}$-free graph of order $n$ with chromatic number at least $s$ that maximizes the spectral radius. Nikiforov [Linear Algebra Appl., 2007] proved…

Given a 3-graph H, let \ex_2(n, H) denote the maximum value of the minimum codegree of a 3-graph on n vertices which does not contain a copy of H. Let F denote the Fano plane, which is the 3-graph \{axx',ayy',azz',xyz',xy'z,x'yz,x'y'z'\}.…

We prove that every 3-coloring of the edges of the complete graph on n vertices without a rainbow triangle contains a set of order Omega(n^{1/3}log^2 n) which uses at most two colors, and this bound is tight up to a constant factor. This…

The Grundy number of a graph is the minimum number of colors needed to properly color the graph using the first-fit greedy algorithm regardless of the initial vertex ordering. Computing the Grundy number of a graph is an NP-Hard problem.…

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in…

The Bollob\'as--Nikiforov conjecture asserts that for any graph $G \neq K_n$ with $m$ edges and clique number $\omega(G)$, \[ \lambda_1^2(G) + \lambda_2^2(G) \;\leq\; 2\!\left(1 - \frac{1}{\omega(G)}\right)m, \] where $\lambda_1(G) \geq…

Mantel's theorem is a classical result in extremal graph theory which implies that the maximum number of edges of a triangle-free graph of order $n$. In 1970, E. Nosal obtained a spectral version of Mantel's theorem which gave the maximum…

Let $\chi_{\Delta}(\mathbb{R}^{n})$ denote the minimum number of colors needed to color $\mathbb{R}^{n}$ so that there will not be a monochromatic equilateral triangle with side length $1$. Using the slice rank method, we reprove a result…