Ramsey's Theorem is among the most well-known results in combinatorics. The theorem states that any two-edge-coloring of a sufficiently large complete graph contains a large monochromatic complete subgraph. Indeed, any two-edge-coloring of a complete graph with N=4t−o(t) vertices contains a monochromatic copy of Kt. On the other hand, a probabilistic argument yields that there exists a two-edge-coloring of a complete graph with N=2t/2+o(t) with no monochromatic copy of Kt. Much attention has been paid to improving these classical bounds but only improvements to lower order terms have been obtained so far. A natural question in this setting is to ask whether every two-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of Kt that can be extended in many ways to a monochromatic copy of Kt+1. Specifically, Erdős, Faudree, Rousseau and Schelp in the 1970's asked whether every two-edge-coloring of KN contains a monochromatic copy of Kt that can be extended in at least (1−ok(1))2−tN ways to a monochromatic copy of Kt+1. A random two-edge-coloring of KN witnesses that this would be best possible. While the intuition coming from random constructions can be misleading, for example, a famous construction by Thomason shows the existence of a two-edge-coloring of a complete graph with fewer monochromatic copies of Kt than a random two-edge-coloring, this paper confirms that the intuition coming from a random construction is correct in this case. In particular, the author answers this question of Erdős et al. in the affirmative. The question can be phrased in the language of Ramsey theory as a problem on determining the Ramsey number of book graphs. A book graph B(k)t is a graph obtained from Kt by adding k new vertices and joining each new vertex to all the vertices of Kt. The main result of the paper asserts that every two-edge-coloring of a complete graph with N=2kt+ok(t) vertices contains a monochromatic copy of B(k)t.
{"title":"The Ramsey number of books","authors":"D. Conlon","doi":"10.19086/aic.10808","DOIUrl":"https://doi.org/10.19086/aic.10808","url":null,"abstract":"Ramsey's Theorem is among the most well-known results in combinatorics. The theorem states that any two-edge-coloring of a sufficiently large complete graph contains a large monochromatic complete subgraph. Indeed, any two-edge-coloring of a complete graph with N=4t−o(t) vertices contains a monochromatic copy of Kt. On the other hand, a probabilistic argument yields that there exists a two-edge-coloring of a complete graph with N=2t/2+o(t) with no monochromatic copy of Kt. Much attention has been paid to improving these classical bounds but only improvements to lower order terms have been obtained so far.\u0000\u0000A natural question in this setting is to ask whether every two-edge-coloring of a sufficiently large complete graph contains a monochromatic copy of Kt that can be extended in many ways to a monochromatic copy of Kt+1. Specifically, Erdős, Faudree, Rousseau and Schelp in the 1970's asked whether every two-edge-coloring of KN contains a monochromatic copy of Kt that can be extended in at least (1−ok(1))2−tN ways to a monochromatic copy of Kt+1. A random two-edge-coloring of KN witnesses that this would be best possible. While the intuition coming from random constructions can be misleading, for example, a famous construction by Thomason shows the existence of a two-edge-coloring of a complete graph with fewer monochromatic copies of Kt than a random two-edge-coloring, this paper confirms that the intuition coming from a random construction is correct in this case. In particular, the author answers this question of Erdős et al. in the affirmative. The question can be phrased in the language of Ramsey theory as a problem on determining the Ramsey number of book graphs. A book graph B(k)t is a graph obtained from Kt by adding k new vertices and joining each new vertex to all the vertices of Kt. The main result of the paper asserts that every two-edge-coloring of a complete graph with N=2kt+ok(t) vertices contains a monochromatic copy of B(k)t.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47555157","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Jan Corsten, Louis DeBiasio, Ander Lamaison, R. Lang
Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices. The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero. A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.
{"title":"Upper density of monochromatic infinite paths","authors":"Jan Corsten, Louis DeBiasio, Ander Lamaison, R. Lang","doi":"10.19086/aic.10810","DOIUrl":"https://doi.org/10.19086/aic.10810","url":null,"abstract":"Ramsey Theory investigates the existence of large monochromatic substructures. Unlike the most classical case of monochromatic complete subgraphs, the maximum guaranteed length of a monochromatic path in a two-edge-colored complete graph is well-understood. Gerencsér and Gyárfás in 1967 showed that any two-edge-coloring of a complete graph Kn contains a monochromatic path with ⌊2n/3⌋+1 vertices. The following two-edge-coloring shows that this is the best possible: partition the vertices of Kn into two sets A and B such that |A|=⌊n/3⌋ and |B|=⌈2n/3⌉, and color the edges between A and B red and edges inside each of the sets blue. The longest red path has 2|A|+1 vertices and the longest blue path has |B| vertices.\u0000\u0000The main result of this paper concerns the corresponding problem for countably infinite graphs. To measure the size of a monochromatic subgraph, we associate the vertices with positive integers and consider the lower and the upper density of the vertex set of a monochromatic subgraph. The upper density of a subset A of positive integers is the limit superior of |A∩{1,...,}|/n, and the lower density is the limit inferior. The following example shows that there need not exist a monochromatic path with positive upper density such that its vertices form an increasing sequence: an edge joining vertices i and j is colored red if ⌊log2i⌋≠⌊log2j⌋, and blue otherwise. In particular, the coloring yields blue cliques with 1, 2, 4, 8, etc., vertices mutually joined by red edges. Likewise, there are constructions of two-edge-colorings such that the lower density of every monochromatic path is zero.\u0000\u0000A result of Rado from the 1970's asserts that the vertices of any k-edge-colored countably infinite complete graph can be covered by k monochromatic paths. For a two-edge-colored complete graph on the positive integers, this implies the existence of a monochromatic path with upper density at least 1/2. In 1993, Erdős and Galvin raised the problem of determining the largest c such that every two-edge-coloring of the complete graph on the positive integers contains a monochromatic path with upper density at least c. The authors solve this 25-year-old problem by showing that c=(12+8–√)/17≈0.87226.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-08-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45442930","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
List coloring is a generalization of graph coloring introduced by Erdős, Rubin and Taylor in 1980, which has become extensively studied in graph theory. A graph G is said to be k-choosable, or k-list-colorable, if, for every way of assigning a list (set) of k colors to each vertex of G, it is possible to choose a color from each list in such a way that no two neighboring vertices receive the same color. Note that if the lists are all the same, then this is asking for G to have chromatic number at most k. One might think that the case where all the lists are the same would be the hardest: surely making the lists different should make it easier to ensure that neighboring vertices have different colors. Rather surprisingly, however, this is not the case. A counterexample is provided by the complete bipartite graph K2,4. If the two vertices in the first vertex class are assigned the lists {a,b} and {c,d}, while the vertices in the other vertex class are assigned the lists {a,c}, {a,d}, {b,c} and {b,d}, then it is easy to check that it is not possible to obtain a proper coloring from these lists, so G is not 2-choosable, and yet the chromatic number of G is 2. A famous theorem of Galvin, which solved the so-called Dinitz conjecture, states that the line graph of the complete bipartite graph Kn,n is n-choosable. Equivalently, if each square of an n×n grid is assigned a list of n colors, it is possible to choose a color from each list in such a way that no color appears more than once in any row or column. One can generalize this notion by requiring a choice of not just one color from each list, but some larger number of colors. A graph G is said to be (A,B)-list-colorable if, for every assignment of lists to the vertices of G, each consisting of A colors, there is an assignment of sets of B colors to the vertices such that each vertex is assigned a set that is a subset of its list and the sets assigned to pairs of adjacent vertices are disjoint. (When B=1 this simply says that G is A-choosable.) In this short paper, the authors answer a question that has remained open for almost four decades since it was posed by Erdős, Rubin and Taylor in their seminal paper: if a graph is (A,B)-list-colorable, is it true that it is also (mA,mB)-list-colorable for every m≥1? Quite surprisingly, the answer is again negative - the authors construct a graph that is (4,1)-list-colorable but not (8,2)-list-colorable.
{"title":"A 4-choosable graph that is not (8:2)-choosable","authors":"Zdenvek Dvovr'ak, Xiaolan Hu, Jean-Sébastien Sereni","doi":"10.19086/aic.10811","DOIUrl":"https://doi.org/10.19086/aic.10811","url":null,"abstract":"List coloring is a generalization of graph coloring introduced by Erdős, Rubin and Taylor in 1980, which has become extensively studied in graph theory. A graph G is said to be k-choosable, or k-list-colorable, if, for every way of assigning a list (set) of k colors to each vertex of G, it is possible to choose a color from each list in such a way that no two neighboring vertices receive the same color. Note that if the lists are all the same, then this is asking for G to have chromatic number at most k.\u0000\u0000One might think that the case where all the lists are the same would be the hardest: surely making the lists different should make it easier to ensure that neighboring vertices have different colors. Rather surprisingly, however, this is not the case. A counterexample is provided by the complete bipartite graph K2,4. If the two vertices in the first vertex class are assigned the lists {a,b} and {c,d}, while the vertices in the other vertex class are assigned the lists {a,c}, {a,d}, {b,c} and {b,d}, then it is easy to check that it is not possible to obtain a proper coloring from these lists, so G is not 2-choosable, and yet the chromatic number of G is 2. A famous theorem of Galvin, which solved the so-called Dinitz conjecture, states that the line graph of the complete bipartite graph Kn,n is n-choosable. Equivalently, if each square of an n×n grid is assigned a list of n colors, it is possible to choose a color from each list in such a way that no color appears more than once in any row or column.\u0000\u0000One can generalize this notion by requiring a choice of not just one color from each list, but some larger number of colors. A graph G is said to be (A,B)-list-colorable if, for every assignment of lists to the vertices of G, each consisting of A colors, there is an assignment of sets of B colors to the vertices such that each vertex is assigned a set that is a subset of its list and the sets assigned to pairs of adjacent vertices are disjoint. (When B=1 this simply says that G is A-choosable.) In this short paper, the authors answer a question that has remained open for almost four decades since it was posed by Erdős, Rubin and Taylor in their seminal paper: if a graph is (A,B)-list-colorable, is it true that it is also (mA,mB)-list-colorable for every m≥1? Quite surprisingly, the answer is again negative - the authors construct a graph that is (4,1)-list-colorable but not (8,2)-list-colorable.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44925405","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Marthe Bonamy, Frantisek Kardos, T. Kelly, P. Nelson, Luke Postle
A well-known conjecture of András Gyárfás and David Sumner states that for every positive integer m and every finite tree T there exists k such that all graphs that do not contain the clique Km or an induced copy of T have chromatic number at most k. The conjecture has been proved in many special cases, but the general case has been open for several decades. The main purpose of this paper is to consider a natural analogue of the conjecture for matroids, where it turns out, interestingly, to be false. Matroids are structures that result from abstracting the notion of independent sets in vector spaces: that is, a matroid is a set M together with a nonempty hereditary collection I of subsets deemed to be independent where all maximal independent subsets of every set are equicardinal. They can also be regarded as generalizations of graphs, since if G is any graph and I is the collection of all acyclic subsets of E(G), then the pair (E(G),I) is a matroid. In fact, it is a binary matroid, which means that it can be represented as a subset of a vector space over F2. To do this, we take the space of all formal sums of vertices and represent the edge vw by the sum v+w. A set of edges is easily seen to be acyclic if and only if the corresponding set of sums is linearly independent. There is a natural analogue of an induced subgraph for matroids: an induced restriction of a matroid M is a subset M′ of M with the property that adding any element of M−M′ to M′ produces a matroid with a larger independent set than M′. The natural analogue of a tree with m edges is the matroid Im, where one takes a set of size m and takes all its subsets to be independent. (Note, however, that unlike with graph-theoretic trees there is just one such matroid up to isomorphism for each m.) Every graph can be obtained by deleting edges from a complete graph. Analogously, every binary matroid can be obtained by deleting elements from a finite binary projective geometry, that is, the set of all one-dimensional subspaces in a finite-dimensional vector space over F2. Finally, the analogue of the chromatic number for binary matroids is a quantity known as the critical number introduced by Crapo and Rota, which in the case of a graph G turns out to be ⌈log2(χ(G))⌉ -- that is, roughly the logarithm of its chromatic number. One of the results of the paper is that a binary matroid can fail to contain I3 or the Fano plane F7 (which is the simplest projective geometry) as an induced restriction, but also have arbitrarily large critical number. By contrast, the critical number is at most two if one also excludes the matroid associated with K5 as an induced restriction. The main result of the paper is a structural description of all simple binary matroids that have neither I3 nor F7 as an induced restriction.
András Gyárfás和David Sumner的一个著名猜想指出，对于每一个正整数m和每一个有限树T，都存在k，使得所有不包含团Km或T的诱导副本的图最多都有色数k。该猜想在许多特殊情况下得到了证明，但一般情况已经开放了几十年。本文的主要目的是考虑拟阵猜想的一个自然类似物，有趣的是，它被证明是假的。拟阵是抽象向量空间中独立集概念的结果：也就是说，拟阵是一个集M和一个被认为是独立的子集的非空遗传集合I，其中每个集的所有最大独立子集都是等基数的。它们也可以看作是图的推广，因为如果G是任何图，并且I是E（G）的所有非循环子集的集合，那么对（E（G），I）是拟阵。事实上，它是一个二进制拟阵，这意味着它可以表示为F2上向量空间的子集。为此，我们取顶点的所有形式和的空间，并用和v+w表示边vw。一组边很容易被看作是非循环的，当且仅当相应的和集是线性无关的。对于拟阵的诱导子图有一个自然的类似物：拟阵M的诱导限制是M的子集M′，其性质是将M−M′的任何元素添加到M′上产生具有比M′更大的独立集的拟阵。具有m条边的树的自然类似物是拟阵Im，其中取一组大小为m的树，并取其所有子集为独立的。（然而，请注意，与图论树不同，每个m只有一个同构的拟阵。）每个图都可以通过从完整图中删除边来获得。类似地，每个二元拟阵都可以通过删除有限二元投影几何中的元素来获得，即F2上有限维向量空间中所有一维子空间的集合，在图G的情况下，结果是？log2（χ（G））？——也就是说，大致是它的色数的对数。本文的结果之一是，二元拟阵可以不包含I3或Fano平面F7（这是最简单的投影几何）作为诱导约束，但也可以具有任意大的临界数。相反，如果还排除了与K5相关的拟阵作为诱导限制，则临界数最多为2。本文的主要结果是对所有既没有I3也没有F7作为诱导限制的简单二元拟阵的结构描述。
{"title":"The Structure of Binary Matroids with no Induced Claw or Fano Plane Restriction","authors":"Marthe Bonamy, Frantisek Kardos, T. Kelly, P. Nelson, Luke Postle","doi":"10.19086/aic.10256","DOIUrl":"https://doi.org/10.19086/aic.10256","url":null,"abstract":"A well-known conjecture of András Gyárfás and David Sumner states that for every positive integer m and every finite tree T there exists k such that all graphs that do not contain the clique Km or an induced copy of T have chromatic number at most k. The conjecture has been proved in many special cases, but the general case has been open for several decades.\u0000\u0000The main purpose of this paper is to consider a natural analogue of the conjecture for matroids, where it turns out, interestingly, to be false. Matroids are structures that result from abstracting the notion of independent sets in vector spaces: that is, a matroid is a set M together with a nonempty hereditary collection I of subsets deemed to be independent where all maximal independent subsets of every set are equicardinal. They can also be regarded as generalizations of graphs, since if G is any graph and I is the collection of all acyclic subsets of E(G), then the pair (E(G),I) is a matroid. In fact, it is a binary matroid, which means that it can be represented as a subset of a vector space over F2. To do this, we take the space of all formal sums of vertices and represent the edge vw by the sum v+w. A set of edges is easily seen to be acyclic if and only if the corresponding set of sums is linearly independent.\u0000\u0000There is a natural analogue of an induced subgraph for matroids: an induced restriction of a matroid M is a subset M′ of M with the property that adding any element of M−M′ to M′ produces a matroid with a larger independent set than M′. The natural analogue of a tree with m edges is the matroid Im, where one takes a set of size m and takes all its subsets to be independent. (Note, however, that unlike with graph-theoretic trees there is just one such matroid up to isomorphism for each m.)\u0000\u0000Every graph can be obtained by deleting edges from a complete graph. Analogously, every binary matroid can be obtained by deleting elements from a finite binary projective geometry, that is, the set of all one-dimensional subspaces in a finite-dimensional vector space over F2.\u0000\u0000Finally, the analogue of the chromatic number for binary matroids is a quantity known as the critical number introduced by Crapo and Rota, which in the case of a graph G turns out to be ⌈log2(χ(G))⌉ -- that is, roughly the logarithm of its chromatic number.\u0000\u0000One of the results of the paper is that a binary matroid can fail to contain I3 or the Fano plane F7 (which is the simplest projective geometry) as an induced restriction, but also have arbitrarily large critical number. By contrast, the critical number is at most two if one also excludes the matroid associated with K5 as an induced restriction. The main result of the paper is a structural description of all simple binary matroids that have neither I3 nor F7 as an induced restriction.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43951943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
[Ramsey's Theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large "tame" substructure. The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{alpha_s}$ vertices with $alpha_s<1/2$ for every $sge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2le tle 2s-1$ and $sge 3$.
[拉姆齐定理](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large "tame" substructure.The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{alpha_s}$ vertices with $alpha_s<1/2$ for every $sge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2le tle 2s-1$ and $sge 3$.
{"title":"Improved bounds for the Erdős-Rogers function","authors":"T. Gowers, Oliver Janzer","doi":"10.19086/aic.12048","DOIUrl":"https://doi.org/10.19086/aic.12048","url":null,"abstract":"[Ramsey's Theorem](https://en.wikipedia.org/wiki/Ramsey%27s_theorem) is one of the most prominent results in graph theory. In its simplest form, it asserts that every sufficiently large two-edge-colored complete graph contains a large monochromatic complete subgraph. This theorem has been generalized to a plethora of statements asserting that every sufficiently large structure of a given kind contains a large \"tame\" substructure.\u0000\u0000The article concerns a closely related problem: for a structure with a given property, find a substructure possessing an even stronger property. For example, what is the largest $K_3$-free induced subgraph of an $n$-vertex $K_4$-free graph? The answer to this question is approximately $n^{1/2}$. The lower bound is easy. If a given graph has a vertex of degree at least $n^{1/2}$, then its neighbors induce a $K_3$-free subgraph with at least $n^{1/2}$ vertices. Otherwise, a greedy procedure yields an independent set of size almost $n^{1/2}$. The argument generalizes to $K_s$-free induced subgraphs of $K_{s+1}$-free graphs. Dudek, Retter and Rödl provided a construction showing that the exponent $1/2$ cannot be improved and asked whether the same is the case for $K_s$-free induced subgraphs of $K_{s+2}$-free graphs. The authors answer this question by providing a construction of $K_{s+2}$-free $n$-vertex graphs with no $K_s$-free induced subgraph with $n^{alpha_s}$ vertices with $alpha_s<1/2$ for every $sge 3$. Their arguments extend to the case of $K_t$-free graphs with no large $K_s$-free induced subgraph for $s+2le tle 2s-1$ and $sge 3$.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2018-04-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47989765","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Z. Furedi, T. Jiang, A. Kostochka, D. Mubayi, J. Verstraete
One of the most intruguing conjectures in extremal graph theory is the conjecture of Erdős and Sós from 1962, which asserts that every $n$-vertex graph with more than $frac{k-1}{2}n$ edges contains any $k$-edge tree as a subgraph. Kalai proposed a generalization of this conjecture to hypergraphs. To explain the generalization, we need to define the concept of a tight tree in an $r$-uniform hypergraph, i.e., a hypergraph where each edge contains $r$ vertices. A tight tree is an $r$-uniform hypergraph such that there is an ordering $v_1,ldots,v_n$ of its its vertices with the following property: the vertices $v_1,ldots,v_r$ form an edge and for every $i>r$, there is a single edge $e$ containing the vertex $v_i$ and $r-1$ of the vertices $v_1,ldots,v_{i-1}$, and $esetminus{v_i}$ is a subset of one of the edges consisting only of vertices from $v_1,ldots,v_{i-1}$. The conjecture of Kalai asserts that every $n$-vertex $r$-uniform hypergraph with more than $frac{k-1}{r}binom{n}{r-1}$ edges contains every $k$-edge tight tree as a subhypergraph. The recent breakthrough results on the existence of combinatorial designs by Keevash and by Glock, Kühn, Lo and Osthus show that this conjecture, if true, would be tight for infinitely many values of $n$ for every $r$ and $k$. The article deals with the special case of the conjecture when the sought tight tree is a path, i.e., the edges are the $r$-tuples of consecutive vertices in the above ordering. The case $r=2$ is the famous Erdős-Gallai theorem on the existence of paths in graphs. The case $r=3$ and $k=4$ follows from an earlier work of the authors on the conjecture of Kalai. The main result of the article is the first non-trivial upper bound valid for all $r$ and $k$. The proof is based on techniques developed for a closely related problem where a hypergraph comes with a geometric structure: the vertices are points in the plane in a strictly convex position and the sought path has to zigzag beetwen the vertices.
{"title":"Tight paths in convex geometric hypergraphs","authors":"Z. Furedi, T. Jiang, A. Kostochka, D. Mubayi, J. Verstraete","doi":"10.19086/aic.12044","DOIUrl":"https://doi.org/10.19086/aic.12044","url":null,"abstract":"One of the most intruguing conjectures in extremal graph theory is the conjecture of Erdős and Sós from 1962, which asserts that every $n$-vertex graph with more than $frac{k-1}{2}n$ edges contains any $k$-edge tree as a subgraph. Kalai proposed a generalization of this conjecture to hypergraphs. To explain the generalization, we need to define the concept of a tight tree in an $r$-uniform hypergraph, i.e., a hypergraph where each edge contains $r$ vertices. A tight tree is an $r$-uniform hypergraph such that there is an ordering $v_1,ldots,v_n$ of its its vertices with the following property: the vertices $v_1,ldots,v_r$ form an edge and for every $i>r$, there is a single edge $e$ containing the vertex $v_i$ and $r-1$ of the vertices $v_1,ldots,v_{i-1}$, and $esetminus{v_i}$ is a subset of one of the edges consisting only of vertices from $v_1,ldots,v_{i-1}$. The conjecture of Kalai asserts that every $n$-vertex $r$-uniform hypergraph with more than $frac{k-1}{r}binom{n}{r-1}$ edges contains every $k$-edge tight tree as a subhypergraph. The recent breakthrough results on the existence of combinatorial designs by Keevash and by Glock, Kühn, Lo and Osthus show that this conjecture, if true, would be tight for infinitely many values of $n$ for every $r$ and $k$.\u0000\u0000The article deals with the special case of the conjecture when the sought tight tree is a path, i.e., the edges are the $r$-tuples of consecutive vertices in the above ordering. The case $r=2$ is the famous Erdős-Gallai theorem on the existence of paths in graphs. The case $r=3$ and $k=4$ follows from an earlier work of the authors on the conjecture of Kalai. The main result of the article is the first non-trivial upper bound valid for all $r$ and $k$. The proof is based on techniques developed for a closely related problem where a hypergraph comes with a geometric structure: the vertices are points in the plane in a strictly convex position and the sought path has to zigzag beetwen the vertices.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2017-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48421472","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Peter Allen, Julia Bottcher, Julia Ehrenmuller, A. Taraz
One of the first results in graph theory was Dirac's theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other results all stating that in dense graphs we can find sparse spanning subgraphs. Along these lines, one of the most far-reaching results is the celebrated _Bandwidth Theorem_, proved around 10 years ago by Böttcher, Schacht, and Taraz. It states, rougly speaking, that every $n$-vertex graph with minimum degree at least $left( frac{r-1}{r} + o(1)right) n$ contains a copy of all $n$-vertex graphs $H$ such that $chi(H) leq r$, $Delta (H) = O(1)$, and the bandwidth of $H$ is $o(n)$. This was conjectured earlier by Bollobás and Komlós. The proof is using the Regularity method based on the Regularity Lemma and the Blow-up Lemma. Ever since the Bandwith Theorem came out, it has been open whether one could prove a similar statement for sparse random graphs. In this remarkable, deep paper the authors do just that, they establish sparse random analogues of the Bandwidth Theorem. In particular, the authors show that, for every positive integer $Delta$, if $p gg left(frac{log{n}}{n}right)^{1/Delta}$, then asymptotically almost surely, every subgraph $Gsubseteq G(n, p)$ with $delta(G) geq left( frac{r-1}{r} + o(1)right) np$ contains a copy of every $r$-colourable spanning (i.e., $n$-vertex) graph $H$ with maximum degree at most $Delta$ and bandwidth $o(n)$, provided that $H$ contains at least $C p^{-2}$ vertices that do not lie on a triangle (of $H$). (The requirement about vertices not lying on triangles is necessary, as pointed out by Huang, Lee, and Sudakov.) The main tool used in the proof is the recent monumental sparse Blow-up Lemma due to Allen, Böttcher, Hàn, Kohayakawa, and Person.
{"title":"The Bandwidth Theorem in sparse graphs","authors":"Peter Allen, Julia Bottcher, Julia Ehrenmuller, A. Taraz","doi":"10.19086/aic.12849","DOIUrl":"https://doi.org/10.19086/aic.12849","url":null,"abstract":"One of the first results in graph theory was Dirac's theorem which claims that if the minimum degree in a graph is at least half of the number of vertices, then it contains a Hamiltonian cycle. This result has inspired countless other results all stating that in dense graphs we can find sparse spanning subgraphs. Along these lines, one of the most far-reaching results is the celebrated _Bandwidth Theorem_, proved around 10 years ago by Böttcher, Schacht, and Taraz. It states, rougly speaking, that every $n$-vertex graph with minimum degree at least $left( frac{r-1}{r} + o(1)right) n$ contains a copy of all $n$-vertex graphs $H$ such that $chi(H) leq r$, $Delta (H) = O(1)$, and the bandwidth of $H$ is $o(n)$. This was conjectured earlier by Bollobás and Komlós. The proof is using the Regularity method based on the Regularity Lemma and the Blow-up Lemma.\u0000\u0000Ever since the Bandwith Theorem came out, it has been open whether one could prove a similar statement for sparse random graphs. In this remarkable, deep paper the authors do just that, they establish sparse random analogues of the Bandwidth Theorem. In particular, the authors show that, for every positive integer $Delta$, if $p gg left(frac{log{n}}{n}right)^{1/Delta}$, then asymptotically almost surely, every subgraph $Gsubseteq G(n, p)$ with $delta(G) geq left( frac{r-1}{r} + o(1)right) np$ contains a copy of every $r$-colourable spanning (i.e., $n$-vertex) graph $H$ with maximum degree at most $Delta$ and bandwidth $o(n)$, provided that $H$ contains at least $C p^{-2}$ vertices that do not lie on a triangle (of $H$). (The requirement about vertices not lying on triangles is necessary, as pointed out by Huang, Lee, and Sudakov.) The main tool used in the proof is the recent monumental sparse Blow-up Lemma due to Allen, Böttcher, Hàn, Kohayakawa, and Person.","PeriodicalId":36338,"journal":{"name":"Advances in Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.0,"publicationDate":"2016-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"68392915","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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