arXiv subjects

Dhruv Mubayi

Publications and source records attributed to Dhruv Mubayi.

At least 73 records · Page 4Linked to original sources

arXiv2019

The feasible region of hypergraphs

Let $\mathcal{F}$ be a family of $r$-uniform hypergraphs. The feasible region $Ω(\mathcal{F})$ of $\mathcal{F}$ is the set of points $(x,y)$ in the unit square such that there exists a sequence of $\mathcal{F}$-free $r$-uniform hypergraphs whose edge density approaches $x$ and whose shadow density approaches $y$. The feasible region provides a lot of combinatorial information, for example, the supremum of $y$ over all $(x,y) \in Ω(\mathcal{F})$ is the Turán density $π(\mathcal{F})$, and $Ω(\emptyset)$ gives the Kruskal-Katona theorem. We undertake a systematic study of $Ω(\mathcal{F})$, and prove that $Ω(\mathcal{F})$ is completely determined by a left-continuous almost everywhere differentiable function; and moreover, there exists an $\mathcal{F}$ for which this function is not continuous. We also extend some old related theorems. For example, we generalize a result of Fisher and Ryan to hypergraphs and extend a classical result of Bollobás by almost completely determining the feasible region for cancellative triple systems.

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arXiv2019

A note on pseudorandom Ramsey graphs

For fixed $s \ge 3$, we prove that if optimal $K_s$-free pseudorandom graphs exist, then the Ramsey number $r(s,t) = t^{s-1+o(1)}$ as $t \rightarrow \infty$. Our method also improves the best lower bounds for $r(C_{\ell},t)$ obtained by Bohman and Keevash from the random $C_{\ell}$-free process by polylogarithmic factors for all odd $\ell \geq 5$ and $\ell \in \{6,10\}$. For $\ell = 4$ it matches their lower bound from the $C_4$-free process. We also prove, via a different approach, that $r(C_5, t)> (1+o(1))t^{11/8}$ and $r(C_7, t)> (1+o(1))t^{11/9}$. These improve the exponent of $t$ in the previous best results and appear to be the first examples of graphs $F$ with cycles for which such an improvement of the exponent for $r(F, t)$ is shown over the bounds given by the random $F$-free process and random graphs.

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arXiv2019

Extremal problems for convex geometric hypergraphs and ordered hypergraphs

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arXiv2019

Maximum $\mathcal H$-free subgraphs

Given a family of hypergraphs $\mathcal H$, let $f(m,\mathcal H)$ denote the largest size of an $\mathcal H$-free subgraph that one is guaranteed to find in every hypergraph with $m$ edges. This function was first introduced by Erdős and Komlós in 1969 in the context of union-free families, and various other special cases have been extensively studied since then. In an attempt to develop a general theory for these questions, we consider the following basic issue: which sequences of hypergraph families $\{\mathcal H_m\}$ have bounded $f(m,\mathcal H_m)$ as $m\to\infty$? A variety of bounds for $f(m,\mathcal H_m)$ are obtained which answer this question in some cases. Obtaining a complete description of sequences $\{\mathcal H_m\}$ for which $f(m,\mathcal H_m)$ is bounded seems hopeless.

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arXiv2019

Hypergraphs not containing a tight tree with a bounded trunk ~II: 3-trees with a trunk of size 2

A tight $r$-tree $T$ is an $r$-uniform hypergraph that has an edge-ordering $e_1, e_2, \dots, e_t$ such that for each $i\geq 2$, $e_i$ has a vertex $v_i$ that does not belong to any previous edge and $e_i-v_i$ is contained in $e_j$ for some $j<i$. Kalai conjectured in 1984 that every $n$-vertex $r$-uniform hypergraph with more than $\frac{t-1}{r}\binom{n}{r-1}$ edges contains every tight $r$-tree $T$ with $t$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree $T'$ of $T$ such that vertices in $V(T)\setminus V(T')$ are leaves in $T$. Kalai's Conjecture was proved in 1987 for tight $r$-trees that have a trunk of size one. In a previous paper we proved an asymptotic version of Kalai's Conjecture for all tight $r$-trees that have a trunk of bounded size. In this paper we continue that work to establish the exact form of Kalai's Conjecture for all tight $3$-trees with at least $20$ edges that have a trunk of size two.

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arXiv2019

Ordered and convex geometric trees with linear extremal function

The extremal functions $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ for ordered and convex geometric acyclic graphs $F$ have been extensively investigated by a number of researchers. Basic questions are to determine when $ex_{\rightarrow}(n,F)$ and $ex_{\cir}(n,F)$ are linear in $n$, the latter posed by Braß-Károlyi-Valtr in 2003. In this paper, we answer both these questions for every tree $F$. We give a forbidden subgraph characterization for a family $\cal T$ of ordered trees with $k$ edges, and show that $ex_{\rightarrow}(n,T) = (k - 1)n - {k \choose 2}$ for all $n \geq k + 1$ when $T \in {\cal T}$ and $ex_{\rightarrow}(n,T) = Ω(n\log n)$ for $T \not\in {\cal T}$. We also describe the family of the convex geometric trees with linear Tur\' an number and show that for every convex geometric tree $F$ not in this family, $ex_{\cir}(n,F)= Ω(n\log \log n)$.

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arXiv2018

The Erdos-Szekeres problem and an induced Ramsey question

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 $c = c(k)$.

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arXiv2018

Extremal problems on ordered and convex geometric hypergraphs

An ordered hypergraph is a hypergraph whose vertex set is linearly ordered, and a convex geometric hypergraph is a hypergraph whose vertex set is cyclically ordered. Extremal problems for ordered and convex geometric graphs have a rich history with applications to a variety of problems in combinatorial geometry. In this paper, we consider analogous extremal problems for uniform hypergraphs, and discover a general partitioning phenomenon which allows us to determine the order of magnitude of the extremal function for various ordered and convex geometric hypergraphs. A special case is the ordered $n$-vertex $r$-graph $F$ consisting of two disjoint sets $e$ and $f$ whose vertices alternate in the ordering. We show that for all $n \geq 2r + 1$, the maximum number of edges in an ordered $n$-vertex $r$-graph not containing $F$ is exactly \[ {n \choose r} - {n - r \choose r}.\] This could be considered as an ordered version of the Erdős-Ko-Rado Theorem, and generalizes earlier results of Capoyleas and Pach and Aronov-Dujmovič-Morin-Ooms-da Silveira.

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arXiv2018

A survey of hypergraph Ramsey problems

The classical hypergraph Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We survey a variety of problems and results in hypergraph Ramsey theory that have grown out of understanding the quantitative aspects of $r_k(s,n)$. Our focus is on recent developments and open problems.

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arXiv2018

Tight paths in convex geometric hypergraphs

In this paper, we prove a theorem on tight paths in convex geometric hypergraphs, which is asymptotically sharp in infinitely many cases. Our geometric theorem is a common generalization of early results of Hopf and Pannwitz, Sutherland, Kupitz and Perles for convex geometric graphs, as well as the classical Erdős-Gallai Theorem for graphs. As a consequence, we obtain the first substantial improvement on the Turán problem for tight paths in uniform hypergraphs.

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arXiv2018

New lower bounds for hypergraph Ramsey numbers

The Ramsey number $r_k(s,n)$ is the minimum $N$ such that for every red-blue coloring of the $k$-tuples of $\{1,\ldots, N\}$, there are $s$ integers such that every $k$-tuple among them is red, or $n$ integers such that every $k$-tuple among them is blue. We prove the following new lower bounds for 4-uniform hypergraph Ramsey numbers: $$r_4(5,n) > 2^{n^{c\log n}} \qquad \hbox{ and } \qquad r_4(6,n) > 2^{2^{cn^{1/5}}},$$ where $c$ is an absolute positive constant. This substantially improves the previous best bounds of $2^{n^{c\log\log n}}$ and $2^{n^{c\log n}}$, respectively. Using previously known upper bounds, our result implies that the growth rate of $r_4(6,n)$ is double exponential in a power of $n$. As a consequence, we obtain similar bounds for the $k$-uniform Ramsey numbers $r_k(k+1, n)$ and $r_k(k+2, n)$ where the exponent is replaced by an appropriate tower function. This almost solves the question of determining the tower growth rate for {\emph {all}} classical off-diagonal hypergraph Ramsey numbers, a question first posed by Erd\H os and Hajnal in 1972. The only problem that remains is to prove that $r_4(5,n)$ is double exponential in a power of $n$.

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arXiv2018

Independence number of graphs with a prescribed number of cliques

We consider the following problem posed by Erdos in 1962. Suppose that $G$ is an $n$-vertex graph where the number of $s$-cliques in $G$ is $t$. How small can the independence number of $G$ be? Our main result suggests that for fixed $s$, the smallest possible independence number undergoes a transition at $t=n^{s/2+o(1)}$. In the case of triangles ($s=3$) we obtain the following result which is sharp apart from constant factors and generalizes basic results in Ramsey theory: there exists $c>0$ such that every $n$-vertex graph with $t$ triangles has independence number at least $$c \cdot \min\left\{ \sqrt {n \log n}\, , \, \frac{n}{t^{1/3}} \left(\log \frac{n}{ t^{1/3}}\right)^{2/3} \right\}.$$

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arXiv2018

Constructions in Ramsey theory

We provide several constructions for problems in Ramsey theory. First, we prove a superexponential lower bound for the classical 4-uniform Ramsey number $r_4(5,n)$, and the same for the iterated $(k-4)$-fold logarithm of the $k$-uniform version $r_k(k+1,n)$. This is the first improvement of the original exponential lower bound for $r_4(5,n)$ implicit in work of Erd\H os and Hajnal from 1972 and also improves the current best known bounds for larger $k$ due to the authors. Second, we prove an upper bound for the hypergraph Erd\H os-Rogers function $f^k_{k+1, k+2}(N)$ that is an iterated $(k-13)$-fold logarithm in $N$. This improves the previous upper bounds that were only logarithmic and addresses a question of Dudek and the first author that was reiterated by Conlon, Fox and Sudakov. Third, we generalize the results of Erd\H os and Hajnal about the 3-uniform Ramsey number of $K_4$ minus an edge versus a clique to $k$-uniform hypergraphs.

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arXiv2017

Hypergraphs not containing a tight tree with a bounded trunk

An $r$-uniform hypergraph is a tight $r$-tree if its edges can be ordered so that every edge $e$ contains a vertex $v$ that does not belong to any preceding edge and the set $e-v$ lies in some preceding edge. A conjecture of Kalai [Kalai], generalizing the Erdős-Sós Conjecture for trees, asserts that if $T$ is a tight $r$-tree with $t$ edges and $G$ is an $n$-vertex $r$-uniform hypergraph containing no copy of $T$ then $G$ has at most $\frac{t-1}{r}\binom{n}{r-1}$ edges. A trunk $T'$ of a tight $r$-tree $T$ is a tight subtree such that every edge of $T-T'$ has $r-1$ vertices in some edge of $T'$ and a vertex outside $T'$. For $r\ge 3$, the only nontrivial family of tight $r$-trees for which this conjecture has been proved is the family of $r$-trees with trunk size one in [FF] from 1987. Our main result is an asymptotic version of Kalai's conjecture for all tight trees $T$ of bounded trunk size. This follows from our upper bound on the size of a $T$-free $r$-uniform hypergraph $G$ in terms of the size of its shadow. We also give a short proof of Kalai's conjecture for tight $r$-trees with at most four edges. In particular, for $3$-uniform hypergraphs, our result on the tight path of length $4$ implies the intersection shadow theorem of Katona [Katona].

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arXiv2017

A short proof of a lower bound for Turán numbers

Let $F$ be a strictly balanced $r$-uniform hypergraph with $e>2$ edges and $r$-density $m$. We give a new short proof of the fact that the Turán number $\ex(n, F)$ is greater than $c\, n^{r-1/m} (\log n)^{1/(e-1)}$ where $c$ depends only on $F$. The previous proof of this for $r=2$ by Bohman and Keevash and for $r \ge 3$ by Bennett and Bohman used a random greedy process and its analysis using the differential equations method. Our proof uses elementary probabilistic arguments together with a (nontrivial) classical result about independent sets in hypergraphs.

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arXiv2017

The number of triple systems without even cycles

For $k \ge 4$, a loose $k$-cycle $C_k$ is a hypergraph with distinct edges $e_1, e_2, \ldots, e_k$ such that consecutive edges (modulo $k$) intersect in exactly one vertex and all other pairs of edges are disjoint. Our main result is that for every even integer $k \ge 4$, there exists $c>0$ such that the number of triple systems with vertex set $[n]$ containing no $C_{k}$ is at most $2^{cn^2}$. An easy construction shows that the exponent is sharp in order of magnitude. This may be viewed as a hypergraph extension of the work of Morris and Saxton, who proved the analogous result for graphs which was a longstanding problem. For $r$-uniform hypergraphs with $r>3$, we improve the trivial upper bound but fall short of obtaining the order of magnitude in the exponent, which we conjecture is $n^{r-1}$. Our proof method is different than that used for most recent results of a similar flavor about enumerating discrete structures, since it does not use hypergraph containers. One novel ingredient is the use of some (new) quantitative estimates for an asymmetric version of the bipartite canonical Ramsey theorem.

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arXiv2016

Variants of the Erdos-Szekeres and Erdos-Hajnal Ramsey problems

Given integers $\ell,n$, the $\ell$th power of the path $P_n$ is the ordered graph $P_n^{\ell}$ with vertex set $v_1 1$, Balko-Cibulka-Král-Kynčl proved that $r(P_n^{\ell}, P_n^{\ell})< c_{\ell}n^{128 \ell}$ and asked for the growth rate for fixed $\ell$. When $\ell=2$, we improve this upper bound by proving $r(P_n^{2}, P_n^{2})< cn^{19.5}$. Using this result, we determine the correct tower growth rate of the $k$-uniform hypergraph ramsey number of a $(k+1)$-clique versus an ordered tight path. Finally, we consider an ordered version of the classical Erd Hos-Hajnal hypergraph ramsey problem, improve the tower height given by the trivial upper bound, and conjecture that this tower height is optimal.

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arXiv2016

Extremal theory of locally sparse multigraphs

An $(n,s,q)$-graph is an $n$-vertex multigraph where every set of $s$ vertices spans at most $q$ edges. In this paper, we determine the maximum product of the edge multiplicities in $(n,s,q)$-graphs if the congruence class of $q$ modulo ${s\choose 2}$ is in a certain interval of length about $3s/2$. The smallest case that falls outside this range is $(s,q)=(4,15)$, and here the answer is $a^{n^2+o(n^2)}$ where $a$ is transcendental assuming Schanuel's conjecture. This could indicate the difficulty of solving the problem in full generality. Many of our results can be seen as extending work by Bondy-Tuza and Füredi-Kündgen about sums of edge multiplicities to the product setting. We also prove a variety of other extremal results for $(n,s,q)$-graphs, including product-stability theorems. These results are of additional interest because they can be used to enumerate and to prove logical 0-1 laws for $(n,s,q)$-graphs. Our work therefore extends many classical enumerative results in extremal graph theory beginning with the Erdős-Kleitman-Rothschild theorem to multigraphs.

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