Fix integers $n \ge r \ge 2$. A clique partition of ${[n] \choose r}$ is a collection of proper subsets $A_1, A_2, \ldots, A_t \subset [n]$ such that $\bigcup_i{A_i \choose r}$ is a partition of ${[n] \choose r}$.
Let $\cp(n,r)$ denote the minimum size of a clique partition of
${[n] \choose r}$.
A classical theorem of de Bruijn and Erd\H os states that $\cp(n, 2) = n$. In this paper we study $\cp(n,r)$, and show in general that for each fixed $r \geq 3$, \[ \cp(n,r) \geq (1 + o(1))n^{r/2} \quad \quad \mbox{as}n
\rightarrow \infty.\] We conjecture $\cp(n,r) = (1 + o(1))n^{r/2}$. This conjecture has already been verified (in a very strong sense) for $r = 3$ by Hartman-Mullin-Stinson. We give further evidence of this conjecture by constructing, for each $r \ge 4$, a family of $(1+o(1))n^{r/2}$ subsets of $[n]$ with the following property: no two $r$-sets of $[n]$ are covered more than once and all but $o(n^r)$ of the $r$-sets of $[n]$ are covered.
We also give an absolute lower bound $\cp(n,r) \geq {n \choose r}/{q + r - 1 \choose r}$ when $n = q^2 + q + r - 1$, and for each $r$ characterize the finitely many configurations achieving equality with the lower bound. Finally we note the connection of $\cp(n,r)$ to extremal graph theory, and determine some new asymptotically sharp bounds for the Zarankiewicz problem.
The paper is devoted to a proof of the de Bruijn-Erdös theorem in incidence geometry based on the Ph. Hall's marriage theorem (the theorem about the systems of distinct representatives).
A classical theorem of De Bruijn and Erdős asserts that any noncollinear set of n points in the plane determines at least n distinct lines. We prove that an analogue of this theorem holds for graphs. Restricting our attention to comparability graphs, we obtain a version of the De Bruijn-Erdős theorem for partially ordered sets (posets). Moreover, in this case, we have an improved bound on the number of lines depending on the height of the poset. The extremal configurations are also determined.
A special case of a combinatorial theorem of De Bruijn and Erdos asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces where each nonzero distance equals 1 or 2.
In this article a family of recursive and self-similar matrices is constructed. It is shown that the Plücker matrix of the Isotropic Grassmannian variety is a direct sum of this class of matrices.
A special case of a combinatorial theorem of De Bruijn and Erdos asserts that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvatal suggested a possible generalization of this assertion in metric spaces with appropriately defined lines. We prove this generalization in all metric spaces induced by connected chordal graphs.
We show that a 1969 result of Bouwkamp and de Bruijn on a formal power series expansion can be interpreted as the hypergraph analogue of the fact that every connected graph with n vertices has at least n-1 edges. We explain some of Bouwkamp and de Bruijn's formulas in terms of hypertrees and we use Lagrange inversion to count hypertrees by the number of vertices and the number of edges of a specified size.
The paper is devoted to a somewhat idiosyncratic account of the theorem of de Bruijn-Erdös and Hanani from the combinatorics of finite geometries and its various proofs. Among the proofs discussed are the original proofs by de Bruijn--Erdös and Hanani and few others. It is explained how one of the proofs was discovered, and how another one could have been discovered.
De Bruijn and Erdős proved that every noncollinear set of n points in the plane determines at least n distinct lines. Chen and Chvátal suggested a possible generalization of this theorem in the framework of metric spaces. We provide partial results in this direction.
De Bruijn and Erdős proved that every noncollinear set of n points in the plane determines at least n distinct lines. We suggest a possible generalization of this theorem in the framework of metric spaces and provide partial results on related extremal combinatorial problems.
Motivated by a well-known result in extremal set theory, due to Nicolaas Govert de Bruijn and Paul Erdős, we consider curves in the unit $n$-cube $[0,1]^n$ of the form \[ A=\{(x,f_1(x),\ldots,f_{n-2}(x),α): x\in [0,1]\}, \] where $α$ is a fixed real number in $[0,1]$ and $f_1,\ldots,f_{n-2}$ are injective measurable functions from $[0,1]$ to $[0,1]$. We refer to such a curve $A$ as an $n$-\emph{de~Bruijn-Erdős-set}. Under the additional assumption that all functions $f_i,i=1,\ldots,n-2,$ are piecewise monotone, we show that the Hausdorff dimension of $A$ is at most $1$ as well as that its $1$-…
What is the length of the shortest sequence $S$ of reals so that the set of consecutive $n$-words in $S$ form a covering code for permutations on $\{1,2, >..., n\}$ of radius $R$ ? (The distance between two $n$-words is the number of transpositions needed to have the same order type.) The above problem can be viewed as a special case of finding a De Bruijn covering code for a rooted hypergraph. Each edge of a rooted hypergraph contains a special vertex, called the {\it root} of the edge, and each vertex is the root of a unique edge, called its {\it ball}. A De Bruijn covering code is a subset …
Generalizing the concept of dense hypergraph, we say that a hypergraph is weakly dense, if no k in the half-open interval [2,sqrt(n)) is the degree of more than k^2 vertices. In our main result, we prove the famous Erdos-Faber-Lovasz conjecture when the hypergraph is weakly dense.
In 1965 Erdős conjectured that the number of edges in k-uniform hypergraphs on n vertices in which the largest matching has s edges is maximized for hypergraphs of one of two special types. We settled this conjecture in the affirmative for k=3 and n is large enough.
We use an algebraic method to prove a degree version of the celebrated Erd\H os-Ko-Rado theorem: given $n>2k$, every intersecting $k$-uniform hypergraph $H$ on $n$ vertices contains a vertex that lies on at most $\binom{n-2}{k-2}$ edges. This result can be viewed as a special case of the degree version of a well-known conjecture of Erdős on hypergraph matchings. Improving the work of Bollobás, Daykin, and Erd\H os from 1976, we show that given integers $n, k, s$ with $n\ge 3k^2 s$, every $k$-uniform hypergraph $H$ on $n$ vertices with minimum vertex degree greater than $\binom{n-1}{k-1}-\binom…
Fix integers $n \ge r \ge 2$. A clique partition of ${[n] \choose r}$ is a collection of proper subsets $A_1, A_2, ..., A_t \subset [n]$ such that $\bigcup_i{A_i \choose r}$ is a partition of ${[n] \choose r}$. Clique partitions are related to design theory, coding theory, projective geometry, and extremal combinatorics. Let $\cp(n,r)$ denote the minimum size of a clique partition of ${[n] \choose r}$. A classical theorem of de Bruijn and Erd\H os states that $\cp(n, 2) = n$ and also determines the extremal configurations. In this paper we study $\cp(n,r)$, and show in general that for each fi…
We prove the well-known Brown-Erdős-Sós Conjecture for hypergraphs of large uniformity in the following form: any dense linear $r$-graph $G$ has $k$ edges spanning at most $(r-2)k+3$ vertices, provided the uniformity $r$ of $G$ is large enough given the linear density of $G$, and the number of vertices of $G$ is large enough given $r$ and $k$.
Ramsey, Erdos-Rado, and Conlon-Fox-Sudakov have given proofs of the 3-hypergraph Ramsey Theorem with better and better upper bounds on the 3-hypergraph Ramsey Number. Ramsey and Erdos-Rado also prove the a-hypergraph Ramsey Theorem. Conlon-Fox-Sudakov note that their upper bounds on the 3-hypergraph Ramsey Numbers, together with a recurrence of Erdos-Rado (which was the key to the Erdos-Rado proof), yield improved bounds on the a-hypergraph Ramsey numbers. We present all of these proofs and state explicit bounds for the 2-color case and the c-color case. We give a more detailed analysis of the…
We prove a new upper bound for the minimum $d$-degree threshold for perfect matchings in $k$-uniform hypergraphs when $d<k/2$. As a consequence, this determines exact values of the threshold when $0.42k \le d < k/2$ or when $(k,d)=(12,5)$ or $(17,7)$. Our approach is to give an upper bound on the Erdős Matching Conjecture and convert the result to the minimum $d$-degree setting by an approach of Kühn, Osthus and Townsend. To obtain exact thresholds, we also apply a result of Treglown and Zhao.
A well-known theorem of Erdős and Gallai asserts that a graph with no path of length $k$ contains at most $\frac{1}{2}(k-1)n$ edges. Recently Győri, Katona and Lemons gave an extension of this result to hypergraphs by determining the maximum number of hyperedges in an $r$-uniform hypergraph containing no Berge path of length $k$ for all values of $r$ and $k$ except for $k=r+1$. We settle the remaining case by proving that an $r$-uniform hypergraph with more than $n$ hyperedges must contain a Berge path of length $r+1$.
A set of $n$ points in the Euclidean plane determines at least $n$ distinct lines unless these $n$ points are collinear. In 2006, Chen and Chvátal asked whether the same statement holds true in general metric spaces, where the line determined by points $x$ and $y$ is defined as the set consisting of $x$, $y$, and all points $z$ such that one of the three points $x,y,z$ lies between the other two. The conjecture that it does hold true remains unresolved even in the special case where the metric space arises from a connected undirected graph with unit lengths assigned to edges. We trace its curr…
The long-standing Erdős-Faber-Lovász conjecture states that every $n$-uniform linear hypergaph with $n$ edges has a proper vertex-coloring using $n$ colors. In this paper we propose an algebraic framework to the problem and formulate a corresponding stronger conjecture. Using the Combinatorial Nullstellensatz, we reduce the Erdős-Faber-Lovász conjecture to the existence of non-zero coefficients in certain polynomials. These coefficients are in turn related to the number of orientations with prescribed in-degree sequences of some auxiliary graphs. We prove the existence of certain orientations,…
Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed $\varepsilon >0$ such that if $n=2k+1$ and $p> 1-\varepsilon$, then w.h.p. (that is, with probability tending to 1 as $k\rightarrow \infty$), $\mathcal{H}_k (n,p)$ has the "Erdős-Ko-Rado property." We also mention a similar random version of Sperner's Theorem.
A well-known theorem of de Bruijn and Erdős states that any set of $n$ non-collinear points in the plane determines at least $n$ lines. Chen and Chvátal asked whether an analogous statement holds within the framework of finite metric spaces, with lines defined using the notion of {\em betweenness}. In this paper, we prove that the answer is affirmative for sets of $n$ points in the plane with the $L_1$ metric, provided that no two points share their $x$- or $y$-coordinate. In this case, either there is a line that contains all $n$ points, or $X$ induces at least $n$ distinct lines. If points o…
Extensions of Erdős-Gallai Theorem for general hypergraphs are well studied. In this work, we prove the extension of Erdős-Gallai Theorem for linear hypergraphs. In particular, we show that the number of hyperedges in an $n$-vertex $3$-uniform linear hypergraph, without a Berge path of length $k$ as a subgraph is at most $\frac{(k-1)}{6}n$ for $k\geq 4$.
A set $A$ of vertices in an $r$-uniform hypergraph $\mathcal H$ is covered in $\mathcal H$ if there is some vertex $u\not\in A$ such that, for every $(r-1)$-set $B\subset A$, the set $\{u\}\cup B$ is in $\mathcal H$. Erdos and Moser (1970) determined the minimum number of edges in a graph on $n$ vertices such that every $k$-set is covered. We extend this result to $r$-uniform hypergraphs on sufficiently many vertices, and determine the extremal hypergraphs. We also address the problem for directed graphs.
The Erdős--Hajnal Theorem asserts that non-universal graphs, that is, graphs that do not contain an induced copy of some fixed graph $H$, have homogeneous sets of size significantly larger than one can generally expect to find in a graph. We obtain two results of this flavor in the setting of $r$-uniform hypergraphs. A theorem of Rödl asserts that if an $n$-vertex graph is non-universal then it contains an almost homogeneous set (i.e one with edge density either very close to $0$ or $1$) of size $Ω(n)$. We prove that if a $3$-uniform hypergraph is non-universal then it contains an almost homog…
Let $\mathcal{H}=(V,\mathcal{E})$ be an $r$-uniform hypergraph on $n$ vertices and fix a positive integer $k$ such that $1\le k\le r$. A $k$-\emph{matching} of $\mathcal{H}$ is a collection of edges $\mathcal{M}\subset \mathcal{E}$ such that every subset of $V$ whose cardinality equals $k$ is contained in at most one element of $\mathcal{M}$. The $k$-matching number of $\mathcal{H}$ is the maximum cardinality of a $k$-matching. A well-known problem, posed by Erdős, asks for the maximum number of edges in an $r$-uniform hypergraph under constraints on its $1$-matching number. In this article we…
We show that for every positive epsilon there exist positive delta and n_0 such that every 3-uniform hypergraph on n>=n_0 vertices with the property that every k-vertex subset, where k>=delta*n, induces at least (1/4 + epsilon)*{k \choose 3} edges, contains K4- as a subgraph, where K4- is the 3-uniform hypergraph on 4 vertices with 3 edges. This question was originally raised by Erdos and Sos. The constant 1/4 is the best possible.
The Erdős Matching Conjecture for 4-uniform hypergraphs
Center for Discrete Mathematics, Fuzhou University（福州大学离散数学中心）；School of Mathematical Sciences, University of Science and Technology of China（中国科学技术大学数学科学学院）
Motivated by the Erdos-Faber Lovasz conjecture (EFL) for hypergraphs, we explore relationships between several conjectures on the edge coloring of linear hypergraphs. In particular, we are able to increase the class of hypergraphs for which EFL is true.
The Kneser hypergraph ${\rm KG}^r_{n,k}$ is an $r$-uniform hypergraph with vertex set consisting of all $k$-subsets of $\{1,\ldots,n\}$ and any collection of $r$ vertices forms an edge if their corresponding $k$-sets are pairwise disjoint. The random Kneser hypergraph ${\rm KG}^r_{n,k}(p)$ is a spanning subhypergraph of ${\rm KG}^r_{n,k}$ in which each edge of ${\rm KG}^r_{n,k}$ is retained independently of each other with probability $p$. The independence number of random subgraphs of ${\rm KG}^2_{n,k}$ was recently addressed in a series of works by Bollob{á}s, Narayanan, and Raigorodskii (20…
Towards the Erdős matching conjecture for 4-uniform hypergraphs: stability and applications
Rényi Institute（雷尼研究所）；School of Mathematics and Statistics, Xi’an Jiaotong University（西安交通大学数学与统计学院）；University of Science and Technology of China（中国科学技术大学）；Yau Mathematical Sciences Center, Tsinghua University（清华大学丘成桐数学科学中心）
Motivated by the Erdos-Szekeres convex polytope conjecture in $R^d$, we initiate the study of the following induced Ramsey problem for hypergraphs. Given integers $ n > k \geq 5$, what is the minimum integer $g_k(n)$ such that any $k$-uniform hypergraph on $g_k(n)$ vertices with the property that any set of $k + 1$ vertices induces 0, 2, or 4 edges, contains an independent set of size $n$. Our main result shows that $g_k(n) > 2^{cn^{k-4}}$, where $c = c(k)$.
A linear hypergraph is intersecting if any two different edges have exactly one common vertex and an $n$-quasicluster is an intersecting linear hypergraph with $n$ edges each one containing at most $n$ vertices and every vertex is contained in at least two edges. The Erdös-Faber-Lovász Conjecture states that the chromatic number of any $n$-quasicluster is at most $n$. In the present note we prove the correctness of the conjecture for a new infinite class of $n$-quasiclusters using a specific edge coloring of the complete graph.
The celebrated Brown-Erdős-Sós conjecture states that for every fixed $e$, every $3$-uniform hypergraph with $Ω(n^2)$ edges contains $e$ edges spanned by $e+3$ vertices. Up to this date all the approaches towards resolving this problem relied on highly involved applications of the hypergraph regularity method, and yet they supplied only approximate versions of the conjecture, producing $e$ edges spanned by $e+O(\log e/\log \log e)$ vertices. In this short paper we describe a completely different approach, which reduces the problem to a variant of another well-known conjecture in extremal graph…
In this paper we study conditions which guarantee the existence of perfect matchings and perfect fractional matchings in uniform hypergraphs. We reduce this problem to an old conjecture by Erdős on estimating the maximum number of edges in a hypergraph when the (fractional) matching number is given, which we are able to solve in some special cases using probabilistic techniques. Based on these results, we obtain some general theorems on the minimum $d$-degree ensuring the existence of perfect (fractional) matchings. In particular, we asymptotically determine the minimum vertex degree which gua…
Given integers $2\le t \le k+1 \le n$, let $g_k(t,n)$ be the minimum $N$ such that every red/blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ yields either a $(k+1)$-set containing $t$ red $k$-subsets, or an $n$-set with all of its $k$-subsets blue. Erdős and Hajnal proved in 1972 that for fixed $2\le t \le k$, there are positive constants $c_1$ and $c_2$ such that $$ 2^{c_1 n} < g_k(t, n) < twr_{t-1} (n^{c_2}),$$ where $twr_{t-1}$ is a tower of 2's of height $t-2$. They conjectured that the tower growth rate in the upper bound is correct. Despite decades of work on closely related and s…
In 1973 P. Erdős and L. Lovász noticed that any hypergraph whose edges are pairwise intersecting has chromatic number 2 or 3. In the first case, such hypergraph may have any number of edges. However, Erdős and Lovász proved that in the second case, the number of edges is bounded from above. For example, if a hypergraph is $ n $-uniform, has pairwise intersecting edges, and has chromatic number 3, then the number of its edges does not exceed $ n^n $. Recently D.D. Cherkashin improved this bound (see \cite{Ch}). In this paper, we further improve it in the case when the number of vertices of an $…
In 1965, Motzkin and Straus established a remarkable connection between the order of a maximum clique and the Lagrangian of a graph and provided a new proof of Turán's theorem using the connection. The connection of Lagrangians and Turán densities can be also used to prove the fundamental theorem of Erdös-Stone-Simonovits on Turán densities of graphs. Very recently, the study of Turán densities of non-uniform hypergraphs have been motivated by extremal poset problems. In this paper, we attempt to explore the applications of Lagrangian method in determining Turán densities of non-uniform hyperg…
Using the Filmus-Golubev-Lifshitz method to bound the independence number of a hypergraph, we solve some problems concerning multiply intersecting families with biased measure. Among other results we obtain a stability result of a measure version of the Erdos-Ko-Rado theorem for multiply intersecting families.
The Erdos-Hajnal conjecture states that if a graph on n vertices is H-free, that is, it does not contain an induced copy of a given graph H, then it must contain either a clique or an independent set of size n^{d(H)}, where d(H) > 0 depends only on the graph H. Except for a few special cases, this conjecture remains wide open. However, it is known that a H-free graph must contain a complete or empty bipartite graph with parts of polynomial size. We prove an analogue of this result for 3-uniform hypergraphs, showing that if a 3-uniform hypergraph on n vertices is H-free, for any given H, then i…
A new, constructive proof with a small explicit constant is given to the Erdős-Pyber theorem which says that the edges of a graph on $n$ vertices can be partitioned into complete bipartite subgraphs so that every vertex is covered at most $O(n/\log n)$ times. The theorem is generalized to uniform hypergraphs. Similar bounds with smaller constant value is provided for fractional partitioning both for graphs and for uniform hypergraphs. We show that these latter constants cannot be improved by more than a factor of 1.89 even for fractional covering by arbitrary complete multipartite subgraphs or…
We improve upon the lower bound for 3-colour hypergraph Ramsey numbers, showing, in the 3-uniform case, that \[r_3 (l,l,l) \geq 2^{l^{c \log \log l}}.\] The old bound, due to Erdős and Hajnal, was \[r_3 (l,l,l) \geq 2^{c l^2 \log^2 l}.\]
Numerous problems in extremal hypergraph theory ask to determine the maximal size of a $k$-uniform hypergraph on $n$ vertices that does not contain an `enlarged' copy $H^+$ of a fixed hypergraph $H$. These include well-known problems such as the Erdős-Sós `forbidding one intersection' problem and the Frankl-Füredi `special simplex' problem. We present a general approach to such problems, using a `junta approximation method' that originates from analysis of Boolean functions. We prove that any $H^+$-free hypergraph is essentially contained in a `junta' -- a hypergraph determined by a small numb…
We give a probabilistic construction of a $3$-uniform hypergraph on $N$ vertices with independence number $O(\log N / \log \log N)$ in which there are at most two edges among any four vertices. This bound is tight and solves a longstanding open problem of Erdős and Hajnal in Ramsey theory. We further extend this result to prove tight bounds on various other hypergraph Ramsey numbers.
