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A perfect $H$-tiling in a graph $G$ is a collection of vertex-disjoint copies of a graph $H$ in $G$ that covers all vertices of $G$. Motivated by papers of Bush and Zhao and of Balogh, Treglown, and Wagner, we determine the threshold for the existence of a perfect $K_{h,h}$-tiling of a randomly perturbed bipartite graph with linear minimum degree., Comment: 18 pages, 1 figure

Given a graph $H$, we say a graph $G$ is properly rainbow $H$-saturated if there is a proper edge-coloring of $G$ which contains no rainbow copy of $H$, but adding any edge to $G$ makes such an edge-coloring impossible. The proper rainbow saturation number, denoted $\text{sat}^*(n,H)$, is the minimum number of edges in an $n$-vertex rainbow $H$-saturated graph. We determine the proper rainbow saturation number for paths up to an additive constant and asymptotically determine $\text{sat}^*(n,K_4)$. In addition, we bound $\text{sat}^*(n,H)$ when $H$ is a larger clique, tree of diameter at least 4, or odd cycle.

A classical result of Kleitman determines the maximum number $f(n,s)$ of subsets in a family $\mathcal{F}\subseteq 2^{[n]}$ of sets that do not contain distinct sets $F_1,F_2,\dots,F_s$ that are pairwise disjoint in the case $n\equiv 0,-1$ (mod $s$). Katona and Nagy determined the maximum size of a family of subsets of an $n$-element set that does not contain $A_1,A_2,\dots,A_t,B_1,B_2,\dots,B_t$ with $\bigcup_{i=1}^t A_i$ and $\bigcup_{i=1}^t B_i$ being disjoint. In this paper, we consider the problem of finding the maximum number $vex(n,K_{s\times t})$ in a family $\mathcal{F}\subseteq 2^{[n]}$ without sets $F^1_1,\dots,F^1_t,\dots,F^s_1,\dots,F^s_t$ such that $G_j=\bigcup_{i=1}^tF^j_i$ $j=1,2,\dots,s$ are pairwise disjoint. We determine the asymptotics of $2^n-vex(n,K_{s\times t})$ if $n\equiv -1$ (mod $s$) for all $t$, and if $n\equiv 0$ (mod $s$), $t\ge 3$ and show that in this latter case the asymptotics of the $t=2$ subcase is different from both the $t=1$ and $t\ge 3$ subcases.

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 $\phi: P\rightarrow \mathbb{P}(X)$ such that $\phi(p)\subseteq\phi(q)$ if and only if $p\leq_{P} q$. The family $F$ is induced-$P$-saturated if it is maximal with respect to being induced-$P$-free. If $n=|X|$, then the size of the smallest induced-$P$-saturated family in $\mathbb{P}(X)$ is denoted $sat(n,P)$. The poset $2C_2$ is two disjoint 2-chains (the Hasse diagram is two vertex-disjoint edges) and Keszegh, Lemons, Martin, P\'alv\"olgyi, and Patk\'os proved that $n+2\leq sat(n,2C_2)\leq 2n$ and gave one isomorphism class of an induced-$2C_2$-saturated family that achieves the upper bound. We show that the lower bound can be improved to $3n/2 + 1/2$ by examining the necessary structure of a saturated family. In addition, we provide many examples of induced-$2C_2$-saturated families of size $2n$ in $\mathbb{P}(X)$ where $|X|=n$.

A coloring of the edges of a graph $G$ in which every $K_{1,2}$ is totally multicolored is known as a proper coloring and a coloring of the edges of $G$ in which every $K_{1,2}$ and every $K_{2,2}$ is totally multicolored is called a B-coloring. In this paper, we establish that a planar graph with maximum degree $\Delta$ can be B-colored with $\max\{2\Delta,32\}$ colors. This is best-possible for large $\Delta$ because $K_{2,\Delta}$ requires $2\Delta$ colors. In addition, there is an example with $\Delta=4$ that requires $12$ colors. We also establish that an outerplanar graph with maximum degree $\Delta$ can be B-colored with $\max\{\Delta,6\}$ colors. This is almost best-possible because $\Delta$ colors are necessary and there is an example with $\Delta=4$ that requires $5$ colors., Comment: 15 pages, 4 figures

We call a proper edge coloring of a graph $G$ a B-coloring if every 4-cycle of $G$ is colored with four different colors. Let $q_B(G)$ denote the smallest number of colors needed for a B-coloring of $G$. Motivated by earlier papers on B-colorings, here we consider $q_B(G)$ for planar and outerplanar graphs in terms of the maximum degree $\Delta = \Delta(G)$. We prove that $q_B(G)\le 2\Delta+8$ for planar graphs, $q_B(G)\le 2\Delta$ for bipartite planar graphs and $q_B(G)\le \Delta+1$ for outerplanar graphs with $\Delta \ge 4$. We conjecture that, for $\Delta$ sufficiently large, $q_B(G)\le 2\Delta(G)$ for planar $G$ and $q_B(G)\le \Delta(G)$ for outerplanar $G$., Comment: 15 pages, 4 figures, to appear in Journal of Graph Theory

The Erd\H os Matching Conjecture states that the maximum size $f(n,k,s)$ of a family $\mathcal{F}\subseteq \binom{[n]}{k}$ that does not contain $s$ pairwise disjoint sets is $\max\{|\mathcal{A}_{k,s}|,|\mathcal{B}_{n,k,s}|\}$, where $\mathcal{A}_{k,s}=\binom{[sk-1]}{k}$ and $\mathcal{B}_{n,k,s}=\{B\in \binom{[n]}{k}:B\cap [s-1]\neq \emptyset\}$. The case $s=2$ is simply the Erd\H{o}s-Ko-Rado theorem on intersecting families and is well understood. The case $n=sk$ was settled by Kleitman and the uniqueness of the extremal construction was obtained by Frankl. Most results in this area show that if $k,s$ are fixed and $n$ is large enough, then the conjecture holds true. Exceptions are due to Frankl who proved the conjecture and considered variants for $n\in [sk,sk+c_{s,k}]$ if $s$ is large enough compared to $k$. A recent manuscript by Guo and Lu considers non-trivial families with matching number at most $s$ in a similar range of parameters. In this short note, we are concerned with the case $s\ge 3$ fixed, $k$ tending to infinity and $n\in\{sk,sk+1\}$. For $n=sk$, we show the stability of the unique extremal construction of size $\binom{sk-1}{k}=\frac{s-1}{s}\binom{sk}{k}$ with respect to minimal degree. As a consequence we derive $\lim\limits_{k\rightarrow \infty}\frac{f(sk+1,k,s)}{\binom{sk+1}{k}}<\frac{s-1}{s}-\varepsilon_s$ for some positive constant $\varepsilon_s$ which depends only on $s$.

Given a set $X$, a collection $\mathcal{F} \subset \mathcal{P}(X)$ is said to be $k$-Sperner if it does not contain a chain of length $k+1$ under set inclusion and it is saturated if it is maximal with respect to this probability. Gerbner et al. proved that the smallest saturated $k$-Sperner system contains at least $2^{k/2-1}$ elements, and later, Morrison, Noel, and Scott showed that the smallest such set contains no more than $2^{0.976723k}$ elements. We improve both the upper and lower bounds, showing that the size of the smallest saturated $k$-Sperner system lies between $\sqrt{k}2^{k/2}$ and $2^{0.961471k}$., Comment: 11 pages

How many copies of a fixed odd cycle, $C_{2m+1}$, can a planar graph contain? We answer this question asymptotically for $m\in\{2,3,4\}$ and prove a bound which is tight up to a factor of $3/2$ for all other values of $m$. This extends the prior results of Cox--Martin and Lv et al. on the analogous question for even cycles. Our bounds result from a reduction to the following maximum likelihood question: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ edges sampled independently from $\mu$ form either a cycle or a path?

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is equal to a positive proportion of the number of edges in $Q_n$, $H$ is said to have positive Tur\'an density in a hypercube; otherwise it has zero Tur\'an density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has positive or zero Tur\'an density remains a widely open question for general $H$. In this paper we focus on layered graphs, i.e., graphs that are contained in an edge-layer of some hypercube. Graphs $H$ that are not layered have positive Tur\'an density because one can form an $H$-free subgraph of $Q_n$ consisting of edges of every other layer. For example, a $4$-cycle is not layered and has positive Tur\'an density. However, in general it is not obvious what properties layered graphs have. We give a characterisation of layered graphs in terms of edge-colorings. We show that most non-trivial subdivisions have zero Tur\'an density, extending known results on zero Tur\'an density of even cycles of length at least $12$ and of length $8$. However, we prove that there are cubical graphs of girth $8$ that are not layered and thus having positive Tur\'an density. The cycle of length $10$ remains the only cycle for which it is not known whether its Tur\'an density is positive or not. We prove that $ex(Q_n, C_{10})= \Omega(n2^n/ \log^a n)$, for a constant $a$, showing that the extremal number for a $10$-cycle behaves differently from any other cycle of zero Tur\'an density., Comment: New references added, including ones on the recent result about 10-cycles and high girth non-layered graphs. Replaced Theorem 2 that had a gap in the previous version with Section 6 containing associated results

An edge-colored graph is said to contain a rainbow-$F$ if it contains $F$ as a subgraph and every edge of $F$ is a distinct color. The problem of maximizing edges among $n$-vertex properly edge-colored graphs not containing a rainbow-$F$, known as the rainbow Tur\'an problem, was initiated by Keevash, Mubayi, Sudakov and Verstra\"ete. We investigate a variation of this problem with the additional restriction that the graph is planar, and we denote the corresponding extremal number by $\ex_{\p}^*(n,F)$. In particular, we determine $\ex_{\p}^*(n,P_5)$, where $P_5$ denotes the $5$-vertex path., Comment: 22 pages, 9 figures

We prove the endomorphism conjecture for graded posets whose largest Whitney number is at most 4. In particular, this implies the endomorphism conjecture is true for graded posets of width at most 4., Comment: 10 pages

Gy\'arf\'as conjectured in 2011 that every $r$-edge-colored $K_n$ contains a monochromatic component of bounded ("perhaps three") diameter on at least $n/(r-1)$ vertices. Letzter proved this conjecture with diameter four. In this note we improve the result in the case of $r=3$: We show that in every $3$-edge-coloring of $K_n$ either there is a monochromatic component of diameter at most three on at least $n/2$ vertices or every color class is spanning and has diameter at most four., Comment: 4 pages; Published online at Journal of Graph Theory

For given posets $P$ and $Q$ and an integer $n$, the generalized Tur\'an problem for posets, asks for the maximum number of copies of $Q$ in a $P$-free subset of the $n$-dimensional Boolean lattice, $2^{[n]}$. In this paper, among other results, we show the following: (i) For every $n\geq 5$, the maximum number of $2$-chains in a butterfly-free subfamily of $2^{[n]}$ is $\left\lceil\frac{n}{2}\right\rceil\binom{n}{\lfloor n/2\rfloor}$. (ii) For every fixed $s$, $t$ and $k$, a $K_{s,t}$-free family in $2^{[n]}$ has $O\left(n\binom{n}{\lfloor n/2\rfloor}\right)$ $k$-chains. (iii) For every $n\geq 3$, the maximum number of $2$-chains in an $\textbf{N}$-free family is $\binom{n}{\lfloor n/2\rfloor}$, where $\textbf{N}$ is a poset on 4 distinct elements $\{p_1,p_2,q_1,q_2\}$ for which $p_1 < q_1$, $p_2 < q_1$ and $p_2 < q_2$. (iv) We also prove exact results for the maximum number of $2$-chains in a family that has no $5$-path and asymptotic estimates for the number of $2$-chains in a family with no $6$-path., Comment: 13 pages, 3 figures

Given a hereditary property $\mathcal H$ of graphs and some $p\in[0,1]$, the edit distance function $\operatorname{ed}_{\mathcal H}(p)$ is (asymptotically) the maximum proportion of "edits" (edge-additions plus edge-deletions) necessary to transform any graph of density $p$ into a member of $\mathcal H$. For any fixed $p\in[0,1]$, $\operatorname{ed}_{\mathcal H}(p)$ can be computed from an object known as a colored regularity graph (CRG). This paper is concerned with those points $p\in[0,1]$ for which infinitely many CRGs are required to compute $\operatorname{ed}_{\mathcal H}$ on any open interval containing $p$; such a $p$ is called an accumulation point. We show that, as expected, $p=0$ and $p=1$ are indeed accumulation points for some hereditary properties; we additionally determine the slope of $\operatorname{ed}_{\mathcal H}$ at these two extreme points. Unexpectedly, we construct a hereditary property with an accumulation point at $p=1/4$. Finally, we derive a significant structural property about those CRGs which occur at accumulation points., Comment: 22 pages

For a fixed planar graph $H$, let $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In the case when $H$ is a cycle, the asymptotic value of $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_m)$ is currently known for $m\in\{3,4,5,6,8\}$. In this note, we extend this list by establishing $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{10})\sim(n/5)^5$ and $\operatorname{\mathbf{N}}_{\mathcal{P}}(n,C_{12})\sim(n/6)^6$. We prove this by answering the following question for $m\in\{5,6\}$, which is interesting in its own right: which probability mass $\mu$ on the edges of some clique maximizes the probability that $m$ independent samples from $\mu$ form an $m$-cycle?, Comment: 7 pages, 1 figure

We propose a new approach for defining and searching clusters in graphs that represent real technological or transaction networks. In contrast to the standard way of finding dense parts of a graph, we concentrate on the structure of edges between the clusters, as it is motivated by some earlier observations, e.g. in the structure of networks in ecology and economics and by applications of discrete tomography. Mathematically special colorings and chromatic numbers of graphs are studied., Comment: 15 pages, 3 figures, 1 table

For a planar graph $H$, let $\operatorname{\mathbf{N}}_{\mathcal P}(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. In this paper, we prove that $\operatorname{\mathbf{N}}_{\mathcal P}(n,P_7)\sim{4\over 27}n^4$, $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_6)\sim(n/3)^3$, $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_8)\sim(n/4)^4$ and $\operatorname{\mathbf{N}}_{\mathcal P}(n,K_4\{1\})\sim(n/6)^6$, where $K_4\{1\}$ is the $1$-subdivision of $K_4$. In addition, we obtain significantly improved upper bounds on $\operatorname{\mathbf{N}}_{\mathcal P}(n,P_{2m+1})$ and $\operatorname{\mathbf{N}}_{\mathcal P}(n,C_{2m})$ for $m\geq 4$. For a wide class of graphs $H$, the key technique developed in this paper allows us to bound $\operatorname{\mathbf{N}}_{\mathcal P}(n,H)$ in terms of an optimization problem over weighted graphs., Comment: 30 pages

Given a hereditary property of graphs $\mathcal{H}$ and a $p\in [0,1]$, the edit distance function ${\rm ed}_{\mathcal{H}}(p)$ is asymptotically the maximum proportion of edge-additions plus edge-deletions applied to a graph of edge density $p$ sufficient to ensure that the resulting graph satisfies $\mathcal{H}$. The edit distance function is directly related to other well-studied quantities such as the speed function for $\mathcal{H}$ and the $\mathcal{H}$-chromatic number of a random graph. Let $\mathcal{H}$ be the property of forbidding an Erd\H{o}s-R\'{e}nyi random graph $F\sim \mathbb{G}(n_0,p_0)$, and let $\varphi$ represent the golden ratio. In this paper, we show that if $p_0\in [1-1/\varphi,1/\varphi]$, then a.a.s. as $n_0\to\infty$, \begin{align*} {\rm ed}_{\mathcal{H}}(p) = (1+o(1))\,\frac{2\log n_0}{n_0} \cdot\min\left\{ \frac{p}{-\log(1-p_0)}, \frac{1-p}{-\log p_0} \right\}. \end{align*} Moreover, this holds for $p\in [1/3,2/3]$ for any $p_0\in (0,1)$., Comment: 33 pages, 3 figures

In this note, we fix a graph $H$ and ask into how many vertices can each vertex of a clique of size $n$ can be "split" such that the resulting graph is $H$-free. Formally: A graph is an $(n,k)$-graph if its vertex sets is a pairwise disjoint union of $n$ parts of size at most $k$ each such that there is an edge between any two distinct parts. Let $$ f(n,H) = \min \{k \in \mathbb N : \mbox{there is an $(n,k)$-graph $G$ such that $H\not\subseteq G$}\} . $$ Barbanera and Ueckerdt observed that $f(n, H)=2$ for any graph $H$ that is not bipartite. If a graph $H$ is bipartite and has a well-defined Tur\'an exponent, i.e., ${\rm ex}(n, H) = \Theta(n^r)$ for some $r$, we show that $\Omega (n^{2/r -1}) = f(n, H) = O (n^{2/r-1} \log ^{1/r} n)$. We extend this result to all bipartite graphs for which an upper and a lower Tur\'an exponents do not differ by much. In addition, we prove that $f(n, K_{2,t}) =\Theta(n^{1/3})$ for any fixed $t$., Comment: 12 pages

Let ${\rm ex}_{\mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${\rm ex}_{\mathcal{P}}(n,T,H)$ is the well studied function, the planar Tur\'an number of $H$, denoted by ${\rm ex}_{\mathcal{P}}(n,H)$. The topic of extremal planar graphs was initiated by Dowden (2016). He obtained sharp upper bound for both ${\rm ex}_{\mathcal{P}}(n,C_4)$ and ${\rm ex}_{\mathcal{P}}(n,C_5)$. Later on, Y. Lan, et al. continued this topic and proved that ${\rm ex}_{\mathcal{P}}(n,C_6)\leq \frac{18(n-2)}{7}$. In this paper, we give a sharp upper bound ${\rm ex}_{\mathcal{P}}(n,C_6) \leq \frac{5}{2}n-7$, for all $n\geq 18$, which improves Lan's result. We also pose a conjecture on ${\rm ex}_{\mathcal{P}}(n,C_k)$, for $k\geq 7$., Comment: 27 pages, 17 figures

Let $f(n,H)$ denote the maximum number of copies of $H$ in an $n$-vertex planar graph. The order of magnitude of $f(n,P_k)$, where $P_k$ is a path on $k$ vertices, is $n^{{\lfloor{\frac{k-1}{2}}\rfloor}+1}$. In this paper we determine the asymptotic value of $f(n,P_5)$ and give conjectures for longer paths., Comment: 9 pages, 2 figures

A subfamily $\mathcal{G}\subseteq \mathcal{F}\subseteq 2^{[n]}$ of sets is a non-induced (weak) copy of a poset $P$ in $\mathcal{F}$ if there exists a bijection $i:P\rightarrow \mathcal{G}$ such that $p\le_P q$ implies $i(p)\subseteq i(q)$. In the case where in addition $p\le_P q$ holds if and only if $i(p)\subseteq i(q)$, then $\mathcal{G}$ is an induced (strong) copy of $P$ in $\mathcal{F}$. We consider the minimum number $sat(n,P)$ [resp.\ $sat^*(n,P)$] of sets that a family $\mathcal{F}\subseteq 2^{[n]}$ can have without containing a non-induced [induced] copy of $P$ and being maximal with respect to this property, i.e., the addition of any $G\in 2^{[n]}\setminus \mathcal{F}$ creates a non-induced [induced] copy of $P$. We prove for any finite poset $P$ that $sat(n,P)\le 2^{|P|-2}$, a bound independent of the size $n$ of the ground set. For induced copies of $P$, there is a dichotomy: for any poset $P$ either $sat^*(n,P)\le K_P$ for some constant depending only on $P$ or $sat^*(n,P)\ge \log_2 n$. We classify several posets according to this dichotomy, and also show better upper and lower bounds on $sat(n,P)$ and $sat^*(n,P)$ for specific classes of posets. Our main new tool is a special ordering of the sets based on the colexicographic order. It turns out that if $P$ is given, processing the sets in this order and adding the sets greedily into our family whenever this does not ruin non-induced [induced] $P$-freeness, we tend to get a small size non-induced [induced] $P$-saturating family., Comment: Added appendix to v3 from v1, as it was accidentally missing from v2

Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no copy of $\mathcal{P}$ as an induced subposet but every proper superset of $\mathcal{F}$ contains a copy of $\mathcal{P}$ as an induced subposet. The minimum size of an induced-$\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\operatorname{sat}^*(n,\mathcal{P})$, was first studied by Ferrara et al. (2017). Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove $\operatorname{sat}^*(n,\mathcal{D}_2)\geq\sqrt{n}$, improving upon a logarithmic lower bound. For the antichain with $k+1$ elements, we prove $\operatorname{sat}^*(n,\mathcal{A}_{k+1})\geq (1-o_k(1))\frac{kn}{\log_2 k}$, improving upon a lower bound of $3n-1$ for $k\geq 3$.

The Hall ratio of a graph $G$ is the maximum value of $v(H) / \alpha(H)$ taken over all non-null subgraphs $H$ of $G$. For any graph, the Hall ratio is a lower-bound on its fractional chromatic number. In this note, we present various constructions of graphs whose fractional chromatic number grows much faster than their Hall ratio. This refutes a conjecture of Harris.

A graphic sequence $\pi$ is potentially $H$-graphic if there is some realization of $\pi$ that contains $H$ as a subgraph. The Erd\H{o}s-Jacobson-Lehel problem asks to determine $\sigma(H,n)$, the minimum even integer such that any $n$-term graphic sequence $\pi$ with sum at least $\sigma(H,n)$ is potentially $H$-graphic. The parameter $\sigma(H,n)$ is known as the potential function of $H$, and can be viewed as a degree sequence variant of the classical extremal function ${\rm ex}(n,H)$. Recently, Ferrara, LeSaulnier, Moffatt and Wenger [On the sum necessary to ensure that a degree sequence is potentially $H$-graphic, Combinatorica 36 (2016), 687--702] determined $\sigma(H,n)$ asymptotically for all $H$, which is analogous to the Erd\H{o}s-Stone-Simonovits Theorem that determines ${\rm ex}(n,H)$ asymptotically for nonbipartite $H$. In this paper, we investigate a stability concept for the potential number, inspired by Simonovits' classical result on the stability of the extremal function. We first define a notion of stability for the potential number that is a natural analogue to the stability given by Simonovits. However, under this definition, many families of graphs are not $\sigma$-stable, establishing a stark contrast between the extremal and potential functions. We then give a sufficient condition for a graph $H$ to be stable with respect to the potential function, and characterize the stability of those graphs $H$ that contain an induced subgraph of order $\alpha(H)+1$ with exactly one edge., Comment: 20 pages, 3 figures

The dimension of a partially-ordered set (poset), introduced by Dushnik and Miller (1941), has been studied extensively in the literature. Recently, Ueckerdt (2016) proposed a variation called local dimension which makes use of partial linear extensions. While local dimension is bounded above by dimension, they can be arbitrarily far apart as the dimension of the standard example is $n$ while its local dimension is only $3$. Hiraguchi (1955) proved that the maximum dimension of a poset of order $n$ is $n/2$. However, we find a very different result for local dimension, proving a bound of $\Theta(n/\log n)$. This follows from connections with covering graphs using difference graphs which are bipartite graphs whose vertices in a single class have nested neighborhoods. We also prove that the local dimension of the $n$-dimensional Boolean lattice is $\Omega(n/\log n)$ and make progress toward resolving a version of the removable pair conjecture for local dimension., Comment: 13 pages, 1 figure

The poset $Y_{k+1, 2}$ consists of $k+2$ distinct elements $x_1$, $x_2$, \dots, $x_{k}$, $y_1$,$y_2$, such that $x_1 \le x_2 \le \dots \le x_{k} \le y_1$,~$y_2$. The poset $Y'_{k+1, 2}$ is the dual of $Y_{k+1, 2}$ Let $\rm{La}^{\sharp}(n,\{Y_{k+1, 2}, Y'_{k+1, 2}\})$ be the size of the largest family $\mathcal{F} \subset 2^{[n]}$ that contains neither $Y_{k+1,2}$ nor $Y'_{k+1,2}$ as an induced subposet. Methuku and Tompkins proved that $\rm{La}^{\sharp}(n, \{Y_{3,2}, Y'_{3,2}\}) = \Sigma(n,2)$ for $n \ge 3$ and they conjectured the generalization that if $k \ge 2$ is an integer and $n \ge k+1$, then $\rm{La}^{\sharp}(n, \{Y_{k+1,2}, Y'_{k+1,2}\}) = \Sigma(n,k)$. In this paper, we introduce a simple discharging approach and prove this conjecture., Comment: 8 pages

We show that for any set $S\subseteq \mathbb{Z}$, $|S|=4$ there exists a 3-coloring of $\mathbb{Z}$ in which every translate of $S$ receives all three colors. This implies that $S$ has a codensity of at most $1/3$, proving a conjecture of Newman [D. J. Newman, Complements of finite sets of integers, Michigan Math. J. 14 (1967) 481--486]. We also consider related questions in $\mathbb{Z}^d$, $d\geq 2$., Comment: 16 pages, improved presentation

Given a poset $P$, a family $F$ of elements in the Boolean lattice is said to be $P$-saturated if (1) $F$ contains no copy of $P$ as a subposet and (2) every proper superset of $F$ contains a copy of $P$ as a subposet. The maximum size of a $P$-saturated family is denoted by $La(n,P)$, which has been studied for a number of choices of $P$. The minimum size of a $P$-saturated family, $sat(n,P)$, was introduced by Gerbner et al. (2013), and parallels the deep literature on the saturation function for graphs. We introduce and study the concept of saturation for induced subposets. As opposed to induced saturation in graphs, the above definition of saturation for posets extends naturally to the induced setting. We give several exact results and a number of bounds on the induced saturation number for several small posets. We also use a transformation to the biclique cover problem to prove a logarithmic lower bound for a rich infinite family of target posets.

If G is a graph and H is a set of subgraphs of G, then an edge-coloring of G is called H-polychromatic if every graph from H gets all colors present in G on its edges. The H-polychromatic number of G, denoted poly_H(G), is the largest number of colors in an H-polychromatic coloring. In this paper, poly_H(G) is determined exactly when G is a complete graph and H is the family of all 1-factors. In addition poly_H(G) is found up to an additive constant term when G is a complete graph and H is the family of all 2-factors, or the family of all Hamiltonian cycles., Comment: 15 pages, 6 figures

Bootstrap percolation is a deterministic cellular automaton in which vertices of a graph~$G$ begin in one of two states, "dormant" or "active". Given a fixed integer $r$, a dormant vertex becomes active if at any stage it has at least $r$ active neighbors, and it remains active for the duration of the process. Given an initial set of active vertices $A$, we say that $G$ $r$-percolates (from $A$) if every vertex in $G$ becomes active after some number of steps. Let $m(G,r)$ denote the minimum size of a set $A$ such that $G$ $r$-percolates from $A$. Bootstrap percolation has been studied in a number of settings, and has applications to both statistical physics and discrete epidemiology. Here, we are concerned with degree-based density conditions that ensure $m(G,2)=2$. In particular, we give an Ore-type degree sum result that states that if a graph $G$ satisfies $\sigma_2(G)\ge n-2$, then either $m(G,2)=2$ or $G$ is in one of a small number of classes of exceptional graphs. We also give a Chv\'{a}tal-type degree condition: If $G$ is a graph with degree sequence $d_1\le d_2\le\dots\le d_n$ such that $d_i \geq i+1$ or $d_{n-i} \geq n-i-1$ for all $1 \leq i < \frac{n}{2}$, then $m(G,2)=2$ or $G$ falls into one of several specific exceptional classes of graphs. Both of these results are inspired by, and extend, an Ore-type result in [D. Freund, M. Poloczek, and D. Reichman, Contagious sets in dense graphs, to appear in European J. Combin.], Comment: 15 pages, 3 figures

The $\mathcal{N}$ poset consists of four distinct sets $W,X,Y,Z$ such that $W\subset X$, $Y\subset X$, and $Y\subset Z$ where $W$ is not necessarily a subset of $Z$. A family $\mathcal{F}$ as a subposet of the $n$-dimensional Boolean lattice, $\mathcal{B}_n$, is $\mathcal{N}$-free if it does not contain $\mathcal{N}$ as a subposet. Let $\text{La}(n, \mathcal{N})$ be the size of a largest $\mathcal{N}$-free family in $\mathcal{B}_n$. Katona and Tarj\'{a}n proved that $\text{La}(n,\mathcal{N})\geq {n \choose k}+A(n,4,k+1)$, where $k=\lfloor n/2\rfloor$ and $A(n, 4, k+1)$ is the size of a single-error-correcting code with constant weight $k+1$. In this note, we prove for $n$ even and $k=n/2$, $\text{La}(n, \mathcal{N}) \geq {n\choose k}+A(n, 4, k)$, which improves the bound on $\text{La}(n, \mathcal{N})$ in the second order term for some values of $n$ and should be an improvement for an infinite family of values of $n$, depending on the behavior of the function $A(n,4,\cdot)$.

Let $c$ be a positive constant. Suppose that $r=o(n^{5/12})$ and the members of $\binom{[n]}{r}$ are chosen sequentially at random to form an intersecting hypergraph $\mathcal{H}$. We show that whp $\mathcal{H}$ consists of a simple hypergraph $\mathcal{S}$ of size $\Theta(r/n^{1/3})$, a distinguished vertex $v$ and all $r$-sets which contain $v$ and meet every edge of $\mathcal{S}$. This is a continuation of the study of such random intersecting systems started in [Electron. J. Combin, (2003) R29] where the case $r=O(n^{1/3})$ was considered. To obtain the stated result we continue to investigate this question in the range $\omega(n^{1/3})\le r \le o(n^{5/12})$., Comment: 20 pages

Let $c$ be a positive constant. We show that if $r=\lfloor cn^{1/3}\rfloor$ and the members of ${[n]\choose r}$ are chosen sequentially at random to form an intersecting hypergraph then with limiting probability $(1+c^3)^{-1}$, as $n\to\infty$, the resulting family will be of maximum size ${n-1\choose r-1}$., Comment: 10 pages in Electron. J. Combin. 10 (2003), Research Paper 29, 10pp

This paper investigates the addition of random edges to arbitrary dense graphs; in particular, we determine the number of random edges required to ensure various monotone properties including the appearance of a fixed size clique, small diameter and $k$-connectivity., Comment: 14 pages

This paper investigates the number of random edges required to add to an arbitrary dense graph in order to make the resulting graph hamiltonian with high probability. Adding $\Theta(n)$ random edges is both necessary and sufficient to ensure this for all such dense graphs. If, however, the original graph contains no large independent set, then many fewer random edges are required. We prove a similar result for directed graphs., Comment: 7 pages, 1 figure

Consider a graph $G$ and a $k$-uniform hypergraph $\mathcal{H}$ on common vertex set $[n]$. We say that $\mathcal{H}$ is $G$-intersecting if for every pair of edges in $X,Y \in \mathcal{H}$ there are vertices $x \in X$ and $y \in Y$ such that $x = y$ or $x$ and $y$ are joined by an edge in $G$. This notion was introduced by Bohman, Frieze, Ruszink\'o and Thoma who proved a natural generalization of the Erd\H{o}s-Ko-Rado Theorem for $G$-intersecting $k$-uniform hypergraphs for $G$ sparse and $k = O( n^{1/4} )$. In this note, we extend this result to $k = O\left( \sqrt{n} \right)$., Comment: 6 pages

Let $G$ be a tripartite graph with $N$ vertices in each vertex class. If each vertex is adjacent to at least $(2/3)N$ vertices in each of the other classes, then either $G$ contains a subgraph that consists of $N$ vertex-disjoint triangles or $G$ is a specific graph in which each vertex is adjacent to exactly $(2/3)N$ vertices in each of the other classes., Comment: 22 pages, 4 figures

Let $G$ be a $d$-regular graph $G$ on $n$ vertices. Suppose that the adjacency matrix of $G$ is such that the eigenvalue $\lambda$ which is second largest in absolute value satisfies $\lambda=o(d)$. Let $G_p$ with $p=\frac{\alpha}{d}$ be obtained from $G$ by including each edge of $G$ independently with probability $p$. We show that if $\alpha<1$ then whp the maximum component size of $G_p$ is $O(\log n)$ and if $\alpha>1$ then $G_p$ contains a unique giant component of size $\Omega(n)$, with all other components of size $O(\log n)$., Comment: 9 pages

This note proves that, given one member, $T$, of a particular family of radius-three trees, every radius-two, triangle-free graph, $G$, with large enough chromatic number contains an induced copy of $T$., Comment: 7 pages, 5 figures