arXiv subjects

Dhruv Mubayi

Publications and source records attributed to Dhruv Mubayi.

At least 91 records · Page 5Linked to original sources

arXiv2016

An extremal graph problem with a transcendental solution

We prove that the number of multigraphs with vertex set $\{1, \ldots, n\}$ such that every four vertices span at most nine edges is $a^{n^2 + o(n^2)}$ where $a$ is transcendental (assuming Schanuel's conjecture from number theory). This is an easy consequence of the solution to a related problem about maximizing the product of the edge multiplicities in certain multigraphs, and appears to be the first explicit (somewhat natural) question in extremal graph theory whose solution is transcendental. These results may shed light on a question of Razborov who asked whether there are conjectures or theorems in extremal combinatorics which cannot be proved by a certain class of finite methods that include Cauchy-Schwarz arguments. Our proof involves a novel application of Zykov symmetrization applied to multigraphs, a rather technical progressive induction, and a straightforward use of hypergraph containers.

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arXiv2016

The Erdős-Hajnal hypergraph Ramsey problem

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 special cases of this problem by many researchers, there have been no improvements of the lower bound for $2<t<k$. Here we settle the Erdős-Hajnal conjecture in almost all cases in a strong form, by determining the correct tower growth rate, and in half of the cases we also determine the correct power of $n$ within the tower. Specifically, we prove that if $2<t<k-1$ and $k - t$ is even, then $$g_k(t, n) = twr_{t-1} (n^{k-t+1 + o(1)}).$$ Similar results are proved for $k - t$ odd.

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arXiv2016

The structure of large intersecting families

A collection of sets is {\em intersecting} if every two members have nonempty intersection. We describe the structure of intersecting families of $r$-sets of an $n$-set whose size is quite a bit smaller than the maximum ${n-1 \choose r-1}$ given by the Erd\H os-Ko-Rado Theorem. In particular, this extends the Hilton-Milner theorem on nontrivial intersecting families and answers a recent question of Han and Kohayakawa for large $n$. In the case $r=3$ we describe the structure of all intersecting families with more than 10 edges. We also prove a stability result for the Erdos matching problem. Our short proofs are simple applications of the Delta-system method introduced and extensively used by Frankl since 1977.

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arXiv2016

Improved bounds for the Ramsey number of tight cycles versus cliques

The 3-uniform tight cycle $C_s^3$ has vertex set $ Z_s$ and edge set $\{\{i, i+1, i+2\}: i \in Z_s\}$. We prove that for every $s \not\equiv 0$ (mod 3) and $s \ge 16$ or $s \in \{8,11,14\}$ there is a $c_s>0$ such that the 3-uniform hypergraph Ramsey number $$r(C_s^3, K_n^3)< 2^{c_s n \log n}$$ This answers in strong form a question of the author and Rödl who asked for an upper bound of the form $2^{n^{1+ε_s}}$ for each fixed $s \ge 4$, where $ε_s \rightarrow 0$ as $s \rightarrow \infty$ and $n$ is sufficiently large. The result is nearly tight as the lower bound is known to be exponential in $n$.

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arXiv2015

Multicolor Sunflowers

A sunflower is a collection of distinct sets such that the intersection of any two of them is the same as the common intersection $C$ of all of them, and $|C|$ is smaller than each of the sets. A longstanding conjecture due to Erdős and Szemerédi states that the maximum size of a family of subsets of $[n]$ that contains no sunflower of fixed size $k>2$ is exponentially smaller than $2^n$ as $n\rightarrow\infty$. We consider this problem for multiple families. In particular, we obtain sharp or almost sharp bounds on the sum and product of $k$ families of subsets of $[n]$ that together contain no sunflower of size $k$ with one set from each family. For the sum, we prove that the maximum is $$(k-1)2^n+1+\sum_{s=n-k+2}^{n}\binom{n}{s}$$ for all $n \ge k \ge 3$, and for the $k=3$ case of the product, we prove that it is between $$\left(\frac{1}{8}+o(1)\right)2^{3n}\qquad \hbox{and} \qquad (0.13075+o(1))2^{3n}.$$

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arXiv2015

The number of trees in a graph

Let $T$ be a tree with $t$ edges. We show that the number of isomorphic (labeled) copies of $T$ in a graph $G = (V,E)$ of minimum degree at least $t$ is at least \[2|E| \prod_{v \in V} (d(v) - t + 1)^{\frac{(t-1)d(v)}{2|E|}}.\] Consequently, any $n$-vertex graph of average degree $d$ and minimum degree at least $t$ contains at least $$nd(d-t+1)^{t-1}$$ isomorphic (labeled) copies of $T$. This answers a question of Dellamonica et. al. (where the above statement was proved when $T$ is the path with three edges) while extending an old result of Erd\H os and Simonovits.

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arXiv2015

Hamilton cycles in quasirandom hypergraphs

We show that, for a natural notion of quasirandomness in $k$-uniform hypergraphs, any quasirandom $k$-uniform hypergraph on $n$ vertices with constant edge density and minimum vertex degree $Ω(n^{k-1})$ contains a loose Hamilton cycle. We also give a construction to show that a $k$-uniform hypergraph satisfying these conditions need not contain a Hamilton $\ell$-cycle if $k-\ell$ divides $k$. The remaining values of $\ell$ form an interesting open question.

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arXiv2015

Off-diagonal hypergraph Ramsey numbers

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ contains a red set of size $s$ or a blue set of size $n$, where a set is red (blue) if all of its $k$-subsets are red (blue). A $k$-uniform \emph{tight path} of size $s$, denoted by $P_{s}$, is a set of $s$ vertices $v_1 < \cdots < v_{s}$ in $\mathbb{Z}$, and all $s-k+1$ edges of the form $\{v_j,v_{j+1},\ldots, v_{j + k -1}\}$. Let $r_k(P_s, n)$ be the minimum $N$ such that every red-blue coloring of the $k$-subsets of $\{1, \ldots, N\}$ results in a red $P_{s}$ or a blue set of size $n$. The problem of estimating both $r_k(s,n)$ and $r_k(P_s, n)$ for $k=2$ goes back to the seminal work of Erdos and Szekeres from 1935, while the case $k\ge 3$ was first investigated by Erdos and Rado in 1952. In this paper, we deduce a quantitative relationship between multicolor variants of $r_k(P_s, n)$ and $r_k(n, n)$. This yields several consequences including the following: (1) We determine the correct tower growth rate for both $r_k(s,n)$ and $r_k(P_s, n)$ for $s \ge k+3$. The question of determining the tower growth rate of $r_k(s,n)$ for all $s \ge k+1$ was posed by Erdos and Hajnal in 1972. (2) We show that determining the tower growth rate of $r_k(P_{k+1}, n)$ is equivalent to determining the tower growth rate of $r_k(n,n)$, which is a notorious conjecture of Erdos, Hajnal and Rado from 1965 that remains open. Some related off-diagonal hypergraph Ramsey problems are also explored.

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arXiv2015

A survey of Turán problems for expansions

The $r$-expansion $G^+$ of a graph $G$ is the $r$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a vertex subset of size $r-2$ disjoint from $V(G)$ such that distinct edges are enlarged by disjoint subsets. Let $ex_r(n,F)$ denote the maximum number of edges in an $r$-uniform hypergraph with $n$ vertices not containing any copy of the $r$-uniform hypergraph $F$. Many problems in extremal set theory ask for the determination of $ex_r(n,G^+)$ for various graphs $G$. We survey these Turán-type problems, focusing on recent developments.

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arXiv2015

On the size-Ramsey number of hypergraphs

The size-Ramsey number of a graph $G$ is the minimum number of edges in a graph $H$ such that every 2-edge-coloring of $H$ yields a monochromatic copy of $G$. Size-Ramsey numbers of graphs have been studied for almost 40 years with particular focus on the case of trees and bounded degree graphs. We initiate the study of size-Ramsey numbers for $k$-uniform hypergraphs. Analogous to the graph case, we consider the size-Ramsey number of cliques, paths, trees, and bounded degree hypergraphs. Our results suggest that size-Ramsey numbers for hypergraphs are extremely difficult to determine, and many open problems remain.

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arXiv2015

Hypergraph Ramsey numbers: tight cycles versus cliques

For $s \ge 4$, the 3-uniform tight cycle $C^3_s$ has vertex set corresponding to $s$ distinct points on a circle and edge set given by the $s$ cyclic intervals of three consecutive points. For fixed $s \ge 4$ and $s \not\equiv 0$ (mod 3) we prove that there are positive constants $a$ and $b$ with $$2^{at} 5$ is proved by using supersaturation and the known upper bound for $r(K_4^{3}, K_t^3)$, while for $s=5$ it follows from a new upper bound for $r(K_5^{3-}, K_t^3)$ that we develop.

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arXiv2015

Discrete metric spaces: structure, enumeration, and $0$-$1$ laws

Fix an integer $r\geq 3$. We consider metric spaces on $n$ points such that the distance between any two points lies in $\{1,..., r\}$. Our main result describes their approximate structure for large $n$. As a consequence, we show that the number of these metric spaces is $\lceil \frac{r+1}{2}\rceil ^{{n\choose 2} + o(n^2)}$. Related results in the continuous setting have recently been proved by Kozma, Meyerovitch, Peled, and Samotij. When $r$ is even, our structural characterization is more precise, and implies that almost all such metric spaces have all distances at least $r/2$. As an easy consequence, when $r$ is even we improve the error term above from $o(n^2)$ to $o(1)$, and also show a labeled first-order $0$-$1$ law in the language $\mathcal{L}_r$, consisting of $r$ binary relations, one for each element of $[r]$. In particular, we show the almost sure theory $T$ is the theory of the Fraïssé limit of the class of all finite simple complete edge-colored graphs with edge colors in $\{r/2,..., r\}$. Our work can be viewed as an extension of a long line of research in extremal combinatorics to the colored setting, as well as an addition to the collection of known structures that admit logical $0$-$1$ laws.

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arXiv2014

Coloring triple systems with local conditions

We produce an edge-coloring of the complete 3-uniform hypergraph on n vertices with $e^{O(\sqrt {log log n})}$ colors such that the edges spanned by every set of five vertices receive at least three distinct colors. This answers the first open case of a question of Conlon-Fox-Lee-Sudakov [1] who asked whether such a coloring exists with $(log n)^{o(1)}$ colors.

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arXiv2014

Eigenvalues of Non-Regular Linear-Quasirandom Hypergraphs

Chung, Graham, and Wilson proved that a graph is quasirandom if and only if there is a large gap between its first and second largest eigenvalue. Recently, the authors extended this characterization to k-uniform hypergraphs, but only for the so-called coregular k-uniform hypergraphs. In this paper, we extend this characterization to all k-uniform hypergraphs, not just the coregular ones. Specifically, we prove that if a k-uniform hypergraph satisfies the correct count of a specially defined four-cycle, then there is a gap between its first and second largest eigenvalue.

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arXiv2014

The independent neighborhoods process

A triangle $T^{(r)}$ in an $r$-uniform hypergraph is a set of $r+1$ edges such that $r$ of them share a common $(r-1)$-set of vertices and the last edge contains the remaining vertex from each of the first $r$ edges. Our main result is that the random greedy triangle-free process on $n$ points terminates in an $r$-uniform hypergraph with independence number $O((n \log n)^{1/r})$. As a consequence, using recent results on independent sets in hypergraphs, the Ramsey number $r(T^{(r)}, K_s^{(r)})$ has order of magnitude $s^r/\log s$. This answers questions posed in~\cite{BFM, KMV} and generalizes the celebrated results of Ajtai-Komlós-Szemerédi~\cite{AKS} and Kim~\cite{K} to hypergraphs.

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arXiv2014

Sparse hypergraphs with low independence number

Let K_4 denote the complete 3-uniform hypergraph on 4 vertices. Ajtai, Erdős, Komlós, and Szemerédi (1981) asked if there is a function ω(d) tending to infinity such that every 3-uniform, K_4-free hypergraph N vertices and average degree d has independence number at least ω(d) N/d^{1/2}. We answer this question by constructing a 3-uniform, K_4-free hypergraph with independence number at most 2N/d^{1/2}. We also provide counterexamples to several related conjectures and improve the lower bound of some hypergraph Ramsey numbers.

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arXiv2014

Turan Problems and Shadows III: expansions of graphs

The expansion $G^+$ of a graph $G$ is the $3$-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a new vertex disjoint from $V(G)$ such that distinct edges are enlarged by distinct vertices. Let $ex_3(n,F)$ denote the maximum number of edges in a $3$-uniform hypergraph with $n$ vertices not containing any copy of a $3$-uniform hypergraph $F$. The study of $ex_3(n,G^+)$ includes some well-researched problems, including the case that $F$ consists of $k$ disjoint edges, $G$ is a triangle, $G$ is a path or cycle, and $G$ is a tree. In this paper we initiate a broader study of the behavior of $ex_3(n,G^+)$. Specifically, we show \[ ex_3(n,K_{s,t}^+) = Θ(n^{3 - 3/s})\] whenever $t > (s - 1)!$ and $s \geq 3$. One of the main open problems is to determine for which graphs $G$ the quantity $ex_3(n,G^+)$ is quadratic in $n$. We show that this occurs when $G$ is any bipartite graph with Turán number $o(n^φ)$ where $φ= \frac{1 + \sqrt{5}}{2}$, and in particular, this shows $ex_3(n,Q^+) = Θ(n^2)$ where $Q$ is the three-dimensional cube graph.

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arXiv2014

Inverse Expander Mixing for Hypergraphs

We formulate and prove inverse mixing lemmas in the settings of simplicial complexes and k-uniform hypergraphs. In the hypergraph setting, we extend results of Bilu and Linial for graphs. In the simplicial complex setting, our results answer a question of Parzanchevski et al.

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