Showing papers in "Combinatorics, Probability & Computing in 2003"
TL;DR: It is proved that, for any fixed bipartite graph H in which all degrees in one colour class are at most r, the Turán number is the maximum possible number of edges in a simple graph on n vertices that contains no copy of H.
Abstract: For a graph $H$ and an integer $n$, the Turan number $\ex(n,H)$ is the maximum possible number of edges in a simple graph on $n$ vertices that contains no copy of $H$. $H$ is $r$-degenerate if every one of its subgraphs contains a vertex of degree at most $r$. We prove that, for any fixed bipartite graph $H$ in which all degrees in one colour class are at most $r$, $\ex(n,H)\,{\leq}\,O(n^{2-1/r})$. This is tight for all values of $r$ and can also be derived from an earlier result of Furedi. We also show that there is an absolute positive constant $c$ such that, for every fixed bipartite $r$-degenerate graph $H$, $\ex(n,H)\,{\leq}\,O(n^{1-c/r}).$ This is motivated by a conjecture of Erdos that asserts that, for every such $H$, $\ex(n,H)\,{\leq}\,O(n^{1-1/r}).$For two graphs $G$ and $H$, the Ramsey number $r(G,H)$ is the minimum number $n$ such that, in any colouring of the edges of the complete graph on $n$ vertices by red and blue, there is either a red copy of $G$ or a blue copy of $H$. Erdos conjectured that there is an absolute constant $c$ such that, for any graph $G$ with $m$ edges, $r(G,G)\,{\leq}\,2^{c \sqrt m}$. Here we prove this conjecture for bipartite graphs $G$, and prove that for general graphs $G$ with $m$ edges, $r(G,G)\,{\leq}\,2^{c \sqrt m \log m}$ for some absolute positive constant $c$.These results and some related ones are derived from a simple and yet surprisingly powerful lemma, proved, using probabilistic techniques, at the beginning of the paper. This lemma is a refined version of earlier results proved and applied by various researchers including Rodl, Kostochka, Gowers and Sudakov.
186 citations
TL;DR: A hunter strategy for general graphs with an escape length of only $\O(n \log (\diam(G)))$ against restricted as well as unrestricted rabbits is found, which is close to optimal since $\Omega(n)$ is a trivial lower bound on the escape length in both models.
Abstract: We analyse a randomized pursuit-evasion game played by two players on a graph, a hunter and a rabbit. Let $G$ be any connected, undirected graph with $n$ nodes. The game is played in rounds and in each round both the hunter and the rabbit are located at a node of the graph. Between rounds both the hunter and the rabbit can stay at the current node or move to another node. The hunter is assumed to be restricted to the graph $G$: in every round, the hunter can move using at most one edge. For the rabbit we investigate two models: in one model the rabbit is restricted to the same graph as the hunter, and in the other model the rabbit is unrestricted, i.e., it can jump to an arbitrary node in every round.We say that the rabbit is caught as soon as hunter and rabbit are located at the same node in a round. The goal of the hunter is to catch the rabbit in as few rounds as possible, whereas the rabbit aims to maximize the number of rounds until it is caught. Given a randomized hunter strategy for $G$, the escape length for that strategy is the worst case expected number of rounds it takes the hunter to catch the rabbit, where the worst case is with regard to all (possibly randomized) rabbit strategies. Our main result is a hunter strategy for general graphs with an escape length of only $\O(n \log (\diam(G)))$ against restricted as well as unrestricted rabbits. This bound is close to optimal since $\Omega(n)$ is a trivial lower bound on the escape length in both models. Furthermore, we prove that our upper bound is optimal up to constant factors against unrestricted rabbits.
96 citations
TL;DR: In the extreme case, when the first round consists of a random graph with edges, where c is a positive constant, it is shown that the Player can win with high probability only if constantly many edges are generated in the second round.
Abstract: We study the following one-person game against a random graph process: the Player's goal is to $2$-colour a random sequence of edges $e_1,e_2,\dots$ of a complete graph on $n$ vertices, avoiding a monochromatic triangle for as long as possible The game is over when a monochromatic triangle is created The online version of the game requires that the Player should colour each edge as it comes, before looking at the next edgeWhile it is not hard to prove that the expected length of this game is about $n^{4/3}$, the proof of the upper bound suggests the following relaxation: instead of colouring online, the random graph is generated in only two rounds, and the Player colours the edges of the first round before the edges of the second round are thrown in Given the size of the first round, how many edges can there be in the second round for the Player to be likely to win? In the extreme case, when the first round consists of a random graph with $cn^{3/2}$ edges, where $c$ is a positive constant, we show that the Player can win with high probability only if constantly many edges are generated in the second round
67 citations
TL;DR: This paper improves the previous bound of to and improves it to for even k, and proves that, which is sharp.
Abstract: Given a positive integer $n$ and a family ${\cal F}$ of graphs, let $f(n,{\cal F})$ denote the maximum number of colours in an edge-colouring of $K_n$ such that no subgraph of $K_n$ belonging to ${\cal F}$ has distinct colours on its edges. Erdos, Simonovits and Sos [6] conjectured for fixed $k$ with $k\geq3$ that $f(n,C_k)\,{=}\, (\frac{k-2}{2}+\frac{1}{k-1})n + O(1)$. This has been proved for $k\leq7$. For general $k$, in this paper we improve the previous bound of $(k-2)n-\big({{k\,{-}\,1}\atop{2}}\big)$ to $f(n,C_k)\leq (\frac{k+1}{2}-\frac{2}{k-1})n - (k-2)$. For even $k$, we further improve it to $\frac{k}{2}n-(k-2)$. We also prove that $f(n,\{C_k,C_{k+1},C_{k+2}\})\leq (\frac{k-2}{2}+\frac{1}{k-1})n-1$, which is sharp.
67 citations
TL;DR: It is shown that, for every , sufficiently large n, and any graph H of order , either H or its complement contains a (d,n)-common graph, that is, a graph in which every set of d vertices has at least n common neighbours.
Abstract: The Ramsey number, $r(G)$, of a graph $G$ is the minimum integer $N$ such that, in every $2$-colouring of the edges of the complete graph $K_N$ on $N$ vertices, there is a monochromatic copy of $G$. In 1975, Burr and Erdos posed a problem on Ramsey numbers of $d$-degenerate graphs, i.e., graphs in which every subgraph has a vertex of degree at most $d$. They conjectured that for every $d$ there exists a constant $c(d)$ such that \[ r(G) \leq c(d)n \] for any $d$-degenerate graph $G$ of order $n$.In this paper we prove that $r(G) \,{\leq}\, n^{1+o(1)}$ for each such $G$. In fact, we show that, for every $\epsilon\,{>}\,0$, sufficiently large $n$, and any graph $H$ of order $n^{1+\epsilon}$, either $H$ or its complement contains a $(d,n)$-common graph, that is, a graph in which every set of $d$ vertices has at least $n$ common neighbours. It is easy to see that any $(d,n)$-common graph contains every $d$-degenerate graph $G$ of order $n$. We further show that, for every constant $C$, there is an $n$ and a graph $H$ of order $Cn$ such that neither $H$ nor its complement contains a $(2,n)$-common graph.
48 citations
TL;DR: It is shown that the largest possible contrast in a secret sharing scheme is approximately $4^{-(k-1)n^k/(n(n-1)\cdots(n-( k-1)))", which implies that thelargest possible contrast equals $4½-1$ in the limit when $n$ approaches infinity.
Abstract: This paper shows that the largest possible contrast $C_{k,n}$ in a $k$-out-of-$n$ secret sharing scheme is approximately $4^{-(k-1)}$. More precisely, we show that $4^{-(k-1)} \leq C_{k,n} \leq 4^{-(k-1)}n^k/(n(n-1)\cdots(n-(k-1)))$. This implies that the largest possible contrast equals $4^{-(k-1)}$ in the limit when $n$ approaches infinity. For large $n$, the above bounds leave almost no gap. For values of $n$ that come close to $k$, we will present alternative bounds (being tight for $n=k$). The proofs of our results proceed by finding a relationship between the largest possible contrast in a secret sharing scheme and the smallest possible approximation error in problems occurring in approximation theory.
45 citations
TL;DR: Non-trivial lower and upper estimates of the value of the control ratio for which the phase transition occurs are established, for any $k > 3$ variables per equation over the finite field GF(2), or equivalently k-XOR-CNF formulas.
Abstract: In this paper we study random linear systems with $k > 3$ variables per equation over the finite field GF(2), or equivalently k-XOR-CNF formulas. In a previous paper Creignou and Daude proved that there exists a phase transition exhibiting a sharp threshold, for the consistency (satisfiability) of such systems (formulas). The control parameter for this transition is the ratio of the number of equations to the number of variables, and the scale for which the transition occurs remains somewhat elusive. In this paper we establish, for any $k > 3$, non-trivial lower and upper estimates of the value of the control ratio for which the phase transition occurs. For $k=3$ we get 0.89 and 0.93, respectively. Moreover, we give experimental results for $k=3$ suggesting that the critical ratio is about 0.92. Our estimates are clearly close to the critical ratio.
40 citations
TL;DR: In this article, the authors consider analogues of van der Waerden's theorem and Szemeredi's theorem, where arithmetic progressions are replaced by binary trees with a fixed distance between successive vertices.
Abstract: We consider analogues of van der Waerden's theorem and Szemeredi's theorem, where arithmetic progressions are replaced by binary trees with a fixed distance between successive vertices. The proofs are based on some novel recurrence properties for Markov processes.
40 citations
TL;DR: It is proved that, for every regular sample graph, if the number of induced copies of L_
u in every induced subgraph is asymptotically the same as in a $p-random graph (up to an error term $o(n^
u)$), then $(G_n)$ is the union of (at most) two quasi- random graph sequences, with possibly distinct attached probabilities.
Abstract: This is a continuation of our work on quasi-random graph properties. The class of quasi-random graphs is defined by certain equivalent graph properties possessed by random graphs. One of the most important of these properties is that, for fixed $
u$, every fixed sample graph $L_
u$ has the same frequency in $G_n$ as in the $p$-random graph. (This holds for both induced and not necessarily induced containment.) In B;9D; we proved that, if the frequency of just one fixed $L_
u$ – as a not necessarily induced subgraph – in every ‘large’ induced subgraph $F_h\subseteq G_n$ is the same as for the random graphs, then $(G_n)$ is quasi-random. Here we shall investigate the analogous problem for induced subgraphs $L_
u$. In such cases $(G_n)$ is not necessarily quasi-random.We shall prove, among other things, that, for every regular sample graph $L_
u$, $
u\geqslant 4$, if the number of induced copies of $L_
u$ in every induced $F_h\subseteq G_n$ is asymptotically the same as in a $p$-random graph (up to an error term $o(n^
u)$), then $(G_n)$ is the union of (at most) two quasi-random graph sequences, with possibly distinct attached probabilities (assuming that $p\in (0,1)$, $e(L_
u)>0$, and $L_
u
e K_
u$).We conjecture the same conclusion for every $L_
u$ with $
u\ge 4$, i.e., even if we drop the assumption of regularity.We shall reduce the general problem to solving a system of polynomials. This gives a ‘simple’ algorithm to decide the problem for every given $L_
u$.
36 citations
TL;DR: For an integer $s\ges 2$, a property $\P^{(s)}$ is an infinite class of s-uniform hypergraphs closed under isomorphism if and only if $\P(s)$ is closed under taking induced subhypergraphs.
Abstract: For an integer $s\ges 2$, a property $\P^{(s)}$ is an infinite class of s-uniform hypergraphs closed under isomorphism. We say that a property $\P^{(s)}$ is \emph{hereditary\/} if~$\P^{(s)}$ is closed under taking induced subhypergraphs. Thus, for some `forbidden class' $\FF=\{\F_i^{(s)}\:i\in I\}$ of s-uniform hypergraphs, $\P^{(s)}$ is the set of all s-uniform hypergraphs not containing any $\F_i^{(s)}\in\FF$ as an induced subhypergraph. Let $\P^{(s)}_n$ be those hypergraphs of $\P^{(s)}$ on some fixed n-vertex set. For a set of s-uniform hypergraphs $\FF=\{\F_i^{(s)}\:i\in I\}$, let\[ \exind(n,\FF)=\max\bigl|[n]^s{\setminus}(\M\cup\N)\vphantom{\big|}\bigr|, \]where the maximum is taken over all $\M$ and $\N\subseteq[n]^s$ with $\M\cap\N=\emptyset$ such that, for all $\G\subseteq[n]^s{\setminus}(\M\cup\N)$, no $\F_i^{(s)}\in\FF$ appears as an induced subhypergraph of $\G\cup\M$. We show that\[ \log_2\big|\P^{(3)}_n\big|=\exind(n,\FF)+o(n^3) \]holds for $s=3$ and any hereditary property $\P^{(3)}$, where $\FF$ is a forbidden class associated with $\P^{(3)}$. This result complements a collection of analogous theorems already proved for graphs (i.e., $s=2$).
35 citations
TL;DR: It is shown that the complete directed graph K can be decomposed as an edge-disjoint union of directed closed trails of lengths m1,…,mt whenever mi[ges ]2 and
Abstract: It has been shown [2] that if n is odd and m1,…,mt are integers with mig3 and ∑i=1t mi=vE(Kn)v then Kn can be decomposed as an edge-disjoint union of closed trails of lengths m1,…,mt. This result was later generalized [3] to all sufficiently dense Eulerian graphs G in place of Kn. In this article we consider the corresponding questions for directed graphs. We show that the complete directed graph ****gif image here**** can be decomposed as an edge-disjoint union of directed closed trails of lengths m1,…,mt whenever mig2 and ****gif image here****, except for the single case when n=6 and all mi=3. We also show that sufficiently dense Eulerian digraphs can be decomposed in a similar manner, and we prove corresponding results for (undirected) complete multigraphs.
TL;DR: The most important general class of (bounded degree, connected) graphs for which it is thought that the equivalence may hold is that of quasi-transitive graphs: it is shown that this is indeed the case.
Abstract: The branching random walk on a regular graph turns out to be particularly easy to analyse using results for the corresponding simple random walk. In this way, one can show that there is an intermediate phase of weak survival if and only if the graph is nonamenable. No such simple analysis holds more generally, and it is known that the nonamenability equivalence does not extend to general connected graphs of bounded degree (although we observe that it does hold for such graphs if the branching random walk is modified in a certain natural way). The most important general class of (bounded degree, connected) graphs for which it is thought that the equivalence may hold is that of quasi-transitive graphs: we show that this is indeed the case.
TL;DR: This work studies properly coloured subgraphs and rainbow sub graphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.
Abstract: A subgraph $H$ in an edge-colouring is properly coloured if incident edges of $H$ are assigned different colours, and $H$ is rainbow if no two edges of $H$ are assigned the same colour. We study properly coloured subgraphs and rainbow subgraphs forced in edge-colourings of complete graphs in which each vertex is incident to a large number of colours.
TL;DR: The minimum number l=l(d, Δ) such that every d-degenerate graph with maximum degree at most Δ admits an equitable t-colouring for every t[ges]l when Δ[ges ]27d.
Abstract: A proper vertex coloring of a graph is called equitable if the sizes of colour classes differ by at most 1. In this paper, we find the minimum number l=l(d, Δ) such that every d-degenerate graph with maximum degree at most Δ admits an equitable t-colouring for every tgl when Δg27d.
TL;DR: Some general results are presented which are intended to be used in further investigations when σ contains relational symbols of arity larger than two or when the set of bounds is infinite.
Abstract: Let $\sigma$ be a finite relational signature, let $\mathcal T$ be a set of finite complete relational structures of signature $\sigma$, and let ${\rm H}_{\mathcal T}$ be the countable homogeneous relational structure of signature $\sigma$ which does not embed any of the structures in $\mathcal T$.When $\sigma$ consists of at most binary relations and $\mathcal T$ is finite, the vertex partition behaviour of ${\rm H}_{\mathcal T}$ is completely analysed, in the sense that it is shown that a canonical partition exists and the size of this partition in terms of the structures in $\mathcal{T}$ is determined. If $\mathcal{T}$ is infinite some results are obtained, but a complete analysis is still missing.Some general results are presented which are intended to be used in further investigations when $\sigma$ contains relational symbols of arity larger than two or when the set of bounds $\mathcal{T}$ is infinite.
TL;DR: Using a functional equation and techniques from complex analysis, the desired properties of this generating function are obtained and the existence of an oscillating factor in this asymptotics is shown rigorously.
Abstract: We investigate the asymptotic behaviour of the average displacement of the simple random walk on the Sierpinski graph. The existence of an oscillating factor in this asymptotics is shown rigorously. The proof depends mainly on the analysis of the corresponding generating function. Using a functional equation and techniques from complex analysis we obtain the desired properties of this generating function.
TL;DR: It is proved that for and 5 the communication complexity of the problem of determining if a word w is of the form , for fixed letters, increases with the length of the word w.
Abstract: We consider the $k$-party communication complexity of the problem of determining if a word $w$ is of the form $w_0a_1w_1a_2\dots w_{k-1}a_kw_k$, for fixed letters $a_1,\dots,a_k$. Using the well-known theorem of Hindman (a Ramsey-type result about finite subsets of natural numbers), we prove that for $k=4$ and $5$ the communication complexity of the problem increases with the length of the word $w$.
TL;DR: The local density version of the Erdős–Stone theorem is obtained and it is proved that the only K_r+1-free graph of order n, in which every αn vertices span at least $\frac{r\,{-}\,1}{2r}(2\alpha -1)n^2$ edges, is a Turán graph.
Abstract: A celebrated theorem of Turan asserts that every graph on n vertices with more than $\frac{r\,{-}\,1}{2r}n^2$ edges contains a copy of a complete graph $K_r+1$. In this paper we consider the following more general question. Let G be a $K_r+1-free graph of order n and let α be a constant, 0<α≤1. How dense can every induced subgraph of G on αn vertices be? We prove the following local density extension of Turan's theorem.For every integer $r\geq 2$ there exists a constant $c_r < 1$ such that, if $c_r < \alpha < 1$ and every αn vertices of G span more than\[ \frac{r\,{-}\,1}{2r}(2\alpha -1)n^2\vspace*{7pt} \]edges, then G contains a copy of $K_r+1$. This result is clearly best possible and answers a question of Erdos, Faudree, Rousseau and Schelp [5].In addition, we prove that the only $K_r+1-free graph of order n, in which every αn vertices span at least $\frac{r\,{-}\,1}{2r}(2\alpha -1)n^2$ edges, is a Turan graph. We also obtain the local density version of the Erdos–Stone theorem.
TL;DR: This year marks the 100th anniversary of the birth of Frank Ramsey and this collection of papers by some of the world’s leading experts, devoted to the theory that bears his name is presented.
Abstract: This year marks the 100th anniversary of the birth of Frank Ramsey. In celebration, Combinatorics, Probability and Computing is delighted to present this collection of papers by some of the world’s leading experts, devoted to the theory that bears his name. During his all too brief life, Ramsey made important contributions to many fields, as indicated in the short biography included here. Apart from this biography, the articles in this double issue are research papers covering many different aspects of Ramsey theory. We would like to thank all the authors, and the anonymous referees, for their contributions to this special issue.
TL;DR: If φ is a scattered order type, and μ is a cardinal, then there exists a scattered orders type ψ such that holds.
Abstract: If $\phi$ is a scattered order type, and $\mu$ is a cardinal, then there exists a scattered order type $\psi$ such that $\psi \to [\phi]^{1}_{\mu,\aleph_0}$ holds.
TL;DR: For a random graph on n vertices where the edges appear with individual rates, this work gives exact formulas for the expected time at which the number of components has gone down to k and the expected length of the corresponding minimal spanning forest.
Abstract: For a random graph on n vertices where the edges appear with individual rates, we give exact formulas for the expected time at which the number of components has gone down to k and the expected length of the corresponding minimal spanning forest.For a random bipartite graph we give a formula for the expected time at which a k-assignment appears. This result has a bearing on the random assignment problem.
TL;DR: It is shown that there are at most $\frac{|V(G)|}{5}$ vertices that are not incident to contractible edges in a 3-connected graph G, and this bound is best-possible.
Abstract: Let G be a simple 3-connected graph with at least five vertices. Tutte [13] showed that G has at least one contractible edge. Thomassen [11] gave a simple proof of this fact and showed that contractible edges have many applications. In this paper, we show that there are at most $\frac{|V(G)|}{5}$ vertices that are not incident to contractible edges in a 3-connected graph G. This bound is best-possible. We also show that if a vertex v is not incident to any contractible edge in G, then v has at least four neighbours having degree three, and each such neighbour is incident to exactly two contractible edges. We give short proofs of several results on contractible edges in 3-connected graphs as well. We also study the contractible elements for k-connected matroids. We partially solve an open problem for regular matroids.
TL;DR: Another construction for low-rank co-diagonal matrices is given, based on a modular sieve formula, of course based on the BBR polynomial of Barrington, Beigel and Rudich.
Abstract: In a previous paper [4] we found a relation between the ranks of co-diagonal matrices (matrices with zeroes in their diagonal and nonzeroes elsewhere) and the quality of explicit Ramsey graph constructions. We also gave a construction based on the BBR polynomial of Barrington, Beigel and Rudich [1]. In the present work we give another construction for low-rank co-diagonal matrices, based on a modular sieve formula.
TL;DR: A version of this result for finite fields is derived from a recent theorem of P. Larick, a short proof of which is also given.
Abstract: A classical result of Sarkozy states that, for any $k\in {\mathbb N}$ and any positive density subset $E$ of ${\mathbb N}$, there exist elements $x$ and $y$ of $E$ and $n
eq 0$ such that $x-y=n^k$. A version of this result for finite fields is derived from a recent theorem of P. Larick, a short proof of which is also given.
TL;DR: A random star-process which begins with n isolated vertices, and in each step chooses randomly a vertex of current minimum degree $\delta$, and connects it with random vertices of degree less than d shows that, for suficiently large d, the resulting final graph is connected with probability 1 - o(1).
Abstract: In this paper we consider a random star $d$-process which begins with $n$ isolated vertices, and in each step chooses randomly a vertex of current minimum degree $\delta$, and connects it with $d - \delta$ random vertices of degree less than $d$. We show that, for $d \geqslant 3$, the resulting final graph is connected with probability $1 - o(1)$, and moreover that, for suficiently large $d$, it is $d$-connected with probability $1 - o(1)$.
TL;DR: It is observed that a combinatorial result of Ruzsa and Szemerédi implies the existence of a 2-Lipschitz subset of size $n^{1/2}\varphi(n)$ in every $n$-point set in $\R^3$, where $\varphi (n) to\infty$ as $n\to-infty$.
Abstract: A set $S\subset \R^d$ is $C$-Lipschitz in the $x_i$-coordinate, where $C>0$ is a real number, if, for every two points $a,b\in S$, we have $|a_i-b_i|\leq C \max\{|a_j-b_j|\sep j=1,2,\ldots,d,\,j
eq i\}$. Motivated by a problem of Laczkovich, the author asked whether every $n$-point set in $\Rd$ contains a subset of size at least $cn^{1-1/d}$ that is $C$-Lipschitz in one of the coordinates, for suitable constants $C$ and $c>0$ (depending on $d$). This was answered negatively by Alberti, Csornyei and Preiss. Here it is observed that a combinatorial result of Ruzsa and Szemeredi implies the existence of a 2-Lipschitz subset of size $n^{1/2}\varphi(n)$ in every $n$-point set in $\R^3$, where $\varphi(n)\to\infty$ as $n\to\infty$.
TL;DR: This paper reformulates the bipartite version of Schütte's well-known tournament problem in terms of intersecting set families and applying probabilistic as well as constructive methods, leading to new open questions.
Abstract: In this paper we consider a bipartite version of Schutte's well-known tournament problem. A bipartite tournament $T=(A,B,E)$ with teams $A$ and $B$, and set of arcs $E$, has the property $S_{k,l}$ if for any subsets $K\subseteq A$ and $L\subseteq B$, with $|K| =k$ and $| L | =l$, there exist conquerors of $K$ and $L$ in opposite teams. The task is to estimate, for fixed $k$ and $l$, the minimum number $f(k,l)=| A | + | B | $ of players in a tournament satisfying property $S_{k,l}$. We achieve this goal by reformulating the problem in terms of intersecting set families and applying probabilistic as well as constructive methods. Intriguing connections with some famous problems of this area have emerged in this way, leading to new open questions.
TL;DR: It is shown that the constant colourings and the one-to-one colourings are insufficient for a canonical version of a certain theorem in Ramsey theory.
Abstract: We show that the constant colourings and the one-to-one colourings are insufficient for a canonical version of a certain theorem in Ramsey theory.
TL;DR: Given a graph H with no isolates, the (generalized) mixed Ramsey number is the smallest integer r such that every H-free graph of order r contains an m-element irredundant set.
Abstract: Given a graph $H$ with no isolates, the (generalized) mixed Ramsey number $r(H, \overline{I}_m)$ is the smallest integer $r$ such that every $H$-free graph of order $r$ contains an $m$-element irredundant set. We consider some questions concerning the asymptotic behaviour of this number (i) with $H$ fixed and $m \rightarrow \infty$, (ii) with $m$ fixed and $(H_n)$ a sequence of dense graphs, in particular for the sequence $(K_n), \; n \rightarrow \infty$. Open problems are mentioned throughout the paper.
TL;DR: A general spectral bound for the sizes of some vertex subsets, which are mutually at a given minimum distance in a graph, is derived and unifies and improves some previous results.
Abstract: A general spectral bound for the sizes of some vertex subsets, which are mutually at a given minimum distance in a graph, is derived. This unifies and improves some previous results. Some applications to the study of certain metric parameters of the graph are then discussed.