arXiv subjects

Dhruv Mubayi

Publications and source records attributed to Dhruv Mubayi.

At least 109 records · Page 6Linked to original sources

arXiv2014

Perfect Packings in Quasirandom Hypergraphs II

For each of the notions of hypergraph quasirandomness that have been studied, we identify a large class of hypergraphs F so that every quasirandom hypergraph H admits a perfect F-packing. An informal statement of a special case of our general result for 3-uniform hypergraphs is as follows. Fix an integer r >= 4 and 0<p<1. Suppose that H is an n-vertex triple system with r|n and the following two properties: * for every graph G with V(G)=V(H), at least p proportion of the triangles in G are also edges of H, * for every vertex x of H, the link graph of x is a quasirandom graph with density at least p. Then H has a perfect $K_r^{(3)}$-packing. Moreover, we show that neither hypotheses above can be weakened, so in this sense our result is tight. A similar conclusion for this special case can be proved by Keevash's hypergraph blowup lemma, with a slightly stronger hypothesis on H.

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arXiv2014

Coloring sparse hypergraphs

Fix $k \geq 3$, and let $G$ be a $k$-uniform hypergraph with maximum degree $Δ$. Suppose that for each $l = 2, ..., k-1$, every set of l vertices of G is in at most $Δ^{(k-l)/(k-1)}/f$ edges. Then the chromatic number of $G$ is $O( (Δ/\log f)^{1/(k-1)})$. This extends results of Frieze and the second author and Bennett and Bohman. A similar result is proved for 3-uniform hypergraphs where every vertex lies in few triangles. This generalizes a result of Alon, Krivelevich, and Sudakov, who proved the result for graphs. Our main new technical contribution is a deviation inequality for positive random variables with expectation less than 1. This may be of independent interest and have further applications.

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arXiv2014

A Ramsey-type result for geometric l-hypergraphs

Let n \geq l \geq 2 and q \geq 2. We consider the minimum N such that whenever we have N points in the plane in general position and the l-subsets of these points are colored with q colors, there is a subset S of n points all of whose l-subsets have the same color and furthermore S is in convex position. This combines two classical areas of intense study over the last 75 years: the Ramsey problem for hypergraphs and the Erd\H os-Szekeres theorem on convex configurations in the plane. For the special case l = 2, we establish a single exponential bound on the minimum N, such that every complete $N$-vertex geometric graph whose edges are colored with q colors, yields a monochromatic convex geometric graph on n vertices. For fixed l \geq 2 and q \geq 4, our results determine the correct exponential tower growth rate for N as a function of n, similar to the usual hypergraph Ramsey problem, even though we require our monochromatic set to be in convex position. Our results also apply to the case of l=3 and q=2 by using a geometric variation of the stepping up lemma of Erd\H os and Hajnal. This is in contrast to the fact that the upper and lower bounds for the usual 3-uniform hypergraph Ramsey problem for two colors differ by one exponential in the tower.

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arXiv2014

Spectral extremal problems for hypergraphs

In this paper we consider spectral extremal problems for hypergraphs. We give two general criteria under which such results may be deduced from `strong stability' forms of the corresponding (pure) extremal results. These results hold for the α-spectral radius defined using the α-norm for any α>1; the usual spectrum is the case α=2. Our results imply that any hypergraph Turán problem which has the stability property and whose extremal construction satisfies some rather mild continuity assumptions admits a corresponding spectral result. A particular example is to determine the maximum α-spectral radius of any 3-uniform hypergraph on n vertices not containing the Fano plane, when n is sufficiently large. Another is to determine the maximum α-spectral radius of any graph on n vertices not containing some fixed colour-critical graph, when n is sufficiently large; this generalizes a theorem of Nikiforov who proved stronger results in the case α=2. We also obtain an α-spectral version of the Erdős-Ko-Rado theorem on t-intersecting k-uniform hypergraphs.

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arXiv2014

Perfect Packings in Quasirandom Hypergraphs

Let k >= 2 and F be a linear k-uniform hypergraph with v vertices. We prove that if n is sufficiently large and v|n, then every quasirandom k-uniform hypergraph on n vertices with constant edge density and minimum degree $Ω(n^{k-1})$ admits a perfect F-packing. The case k = 2 follows immediately from the blowup lemma of Komlós, Sárközy, and Szemerédi. We also prove positive results for some nonlinear F but at the same time give counterexamples for rather simple F that are close to being linear. Finally, we address the case when the density tends to zero, and prove (in analogy with the graph case) that sparse quasirandom 3-uniform hypergraphs admit a perfect matching as long as their second largest eigenvalue is sufficiently smaller than the largest eigenvalue.

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arXiv2014

Turan Problems and Shadows II: Trees

The expansion $G^+$ of a graph $G$ is the 3-uniform hypergraph obtained from $G$ by enlarging each edge of $G$ with a vertex disjoint from $V(G)$ such that distinct edges are enlarged by distinct vertices. Let ex$_r(n,F)$ denote the maximum number of edges in an $r$-uniform hypergraph with $n$ vertices not containing any copy of $F$. The authors \cite{KMV} recently determined ex$_3(n,G^+)$ more generally, namely when $G$ is a path or cycle, thus settling conjectures of Füredi-Jiang \cite{FJ} (for cycles) and Füredi-Jiang-Seiver \cite{FJS} (for paths). Here we continue this project by determining the asymptotics for ex$_3(n,G^+)$ when $G$ is any fixed forest. This settles a conjecture of Füredi \cite{Furedi}. Using our methods, we also show that for any graph $G$, either ex$_3(n,G^{+}) \leq \left(\frac{1}{2} + o(1)\right)n^2$ or ex$_3(n,G^{+}) \geq (1 + o(1))n^2,$ thereby exhibiting a jump for the Turán number of expansions.

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arXiv2013

Counting independent sets in hypergraphs

Let $G$ be a triangle-free graph with $n$ vertices and average degree $t$. We show that $G$ contains at least \[ e^{(1-n^{-1/12})\frac{1}{2}\frac{n}{t}\ln t (\frac{1}{2}\ln t-1)} \] independent sets. This improves a recent result of the first and third authors \cite{countingind}. In particular, it implies that as $n \to \infty$, every triangle-free graph on $n$ vertices has at least $e^{(c_1-o(1)) \sqrt{n} \ln n}$ independent sets, where $c_1 = \sqrt{\ln 2}/4 = 0.208138..$. Further, we show that for all $n$, there exists a triangle-free graph with $n$ vertices which has at most $e^{(c_2+o(1))\sqrt{n}\ln n}$ independent sets, where $c_2 = 1+\ln 2 = 1.693147..$. This disproves a conjecture from \cite{countingind}. Let $H$ be a $(k+1)$-uniform linear hypergraph with $n$ vertices and average degree $t$. We also show that there exists a constant $c_k$ such that the number of independent sets in $H$ is at least \[ e^{c_{k} \frac{n}{t^{1/k}}\ln^{1+1/k}{t}}. \] This is tight apart from the constant $c_k$ and generalizes a result of Duke, Lefmann, and Rödl \cite{uncrowdedrodl}, which guarantees the existence of an independent set of size $Ω(\frac{n}{t^{1/k}} \ln^{1/k}t)$. Both of our lower bounds follow from a more general statement, which applies to hereditary properties of hypergraphs.

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arXiv2013

On the Chromatic Thresholds of Hypergraphs

Let F be a family of r-uniform hypergraphs. The chromatic threshold of F is the infimum of all non-negative reals c such that the subfamily of F comprising hypergraphs H with minimum degree at least $c \binom{|V(H)|}{r-1}$ has bounded chromatic number. This parameter has a long history for graphs (r=2), and in this paper we begin its systematic study for hypergraphs. Łuczak and Thomassé recently proved that the chromatic threshold of the so-called near bipartite graphs is zero, and our main contribution is to generalize this result to r-uniform hypergraphs. For this class of hypergraphs, we also show that the exact Turán number is achieved uniquely by the complete (r+1)-partite hypergraph with nearly equal part sizes. This is one of very few infinite families of nondegenerate hypergraphs whose Turán number is determined exactly. In an attempt to generalize Thomassen's result that the chromatic threshold of triangle-free graphs is 1/3, we prove bounds for the chromatic threshold of the family of 3-uniform hypergraphs not containing {abc, abd, cde}, the so-called generalized triangle. In order to prove upper bounds we introduce the concept of fiber bundles, which can be thought of as a hypergraph analogue of directed graphs. This leads to the notion of fiber bundle dimension, a structural property of fiber bundles that is based on the idea of Vapnik-Chervonenkis dimension in hypergraphs. Our lower bounds follow from explicit constructions, many of which use a hypergraph analogue of the Kneser graph. Using methods from extremal set theory, we prove that these Kneser hypergraphs have unbounded chromatic number. This generalizes a result of Szemerédi for graphs and might be of independent interest. Many open problems remain.

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arXiv2013

The Poset of Hypergraph Quasirandomness

Chung and Graham began the systematic study of k-uniform hypergraph quasirandom properties soon after the foundational results of Thomason and Chung-Graham-Wilson on quasirandom graphs. One feature that became apparent in the early work on k-uniform hypergraph quasirandomness is that properties that are equivalent for graphs are not equivalent for hypergraphs, and thus hypergraphs enjoy a variety of inequivalent quasirandom properties. In the past two decades, there has been an intensive study of these disparate notions of quasirandomness for hypergraphs, and an open problem that has emerged is to determine the relationship between them. Our main result is to determine the poset of implications between these quasirandom properties. This answers a recent question of Chung and continues a project begun by Chung and Graham in their first paper on hypergraph quasirandomness in the early 1990's.

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arXiv2013

On generalized Ramsey numbers for 3-uniform hypergraphs

The well-known Ramsey number $r(t,u)$ is the smallest integer $n$ such that every $K_t$-free graph of order $n$ contains an independent set of size $u$. In other words, it contains a subset of $u$ vertices with no $K_2$. Erd{\H o}s and Rogers introduced a more general problem replacing $K_2$ by $K_s$ for $2\le s<t$. Extending the problem of determining Ramsey numbers they defined the numbers $$ f_{s,t}(n)=\min \big{\{} \max \{|W| : W\subseteq V(G) \text{and} G[W] \text{contains no} K_s\}\big{\}}, $$ where the minimum is taken over all $K_t$-free graphs $G$ of order $n$. In this note, we study an analogous function $f_{s,t}^{(3)}(n)$ for 3-uniform hypergraphs. In particular, we show that there are constants $c_1$ and $c_2$ depending only on $s$ such that $$ c_1(\log n)^{1/4} \left(\frac{\log\log n}{\log\log\log n}\right)^{1/2} < f_{s, s+1}^{(3)}(n) < c_2 \log n. $$

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arXiv2013

Multicolor Ramsey Numbers for Complete Bipartite Versus Complete Graphs

Let H_1, ..., H_k be graphs. The multicolor Ramsey number r(H_1,...,H_k) is the minimum integer r such that in every edge-coloring of K_r by k colors, there is a monochromatic copy of H_i in color i for some 1 <= i <= k. In this paper, we investigate the multicolor Ramsey number $r(K_{2,t},...,K_{2,t},K_m)$, determining the asymptotic behavior up to a polylogarithmic factor for almost all ranges of t and m. Several different constructions are used for the lower bounds, including the random graph and explicit graphs built from finite fields. A technique of Alon and Rödl using the probabilistic method and spectral arguments is employed to supply tight lower bounds. A sample result is $c_1 m^2t/\log^4(mt) \leq r(K_{2,t},K_{2,t},K_m) \leq c_2 m^2t/\log^2 m$ for any t and m, where c_1 and c_2 are absolute constants.

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arXiv2013

Eigenvalues and Linear Quasirandom Hypergraphs

Let p(k) denote the partition function of k. For each k >= 2, we describe a list of p(k)-1 quasirandom properties that a k-uniform hypergraph can have. Our work connects previous notions on linear hypergraph quasirandomness of Kohayakawa-Rödl-Skokan and Conlon-Hàn-Person-Schacht and the spectral approach of Friedman-Wigderson. For each of the quasirandom properties that are described, we define a largest and second largest eigenvalue. We show that a hypergraph satisfies these quasirandom properties if and only if it has a large spectral gap. This answers a question of Conlon-Hàn-Person-Schacht. Our work can be viewed as a partial extension to hypergraphs of the seminal spectral results of Chung-Graham-Wilson for graphs.

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arXiv2013

Turan Problems and Shadows I: Paths and Cycles

A $k$-path is a hypergraph P_k = e_1,e_2,...,e_k such that |e_i \cap e_j| = 1 if |j - i| = 1 and e_i \cap e_j is empty otherwise. A k-cycle is a hypergraph C_k = e_1,e_2,.. ,e_k obtained from a (k-1)-path e_1,e_2,...,e_{k-1} by adding an edge e_k that shares one vertex with e_1, another vertex with e_{k-1} and is disjoint from the other edges. Let ex_r(n,G) be the maximum number of edges in an r-graph with n vertices not containing a given r-graph G. We determine ex_r(n, P_k) and ex_r(n, C_k) exactly for all k \ge 4 and r \ge 3 and $n$ sufficiently large and also characterize the extremal examples. The case k = 3 was settled by Frankl and Füredi. This work is the next step in a long line of research beginning with conjectures of Erd\H os and Sós from the early 1970's. In particular, we extend the work (and settle a recent conjecture) of Füredi, Jiang and Seiver who solved this problem for P_k when r \ge 4 and of Füredi and Jiang who solved it for C_k when r \ge 5. They used the delta system method, while we use a novel approach which involves random sampling from the shadow of an r-graph.

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arXiv2013

Multicolor Ramsey numbers for triple systems

Given an $r$-uniform hypergraph $H$, the multicolor Ramsey number $r_k(H)$ is the minimum $n$ such that every $k$-coloring of the edges of the complete $r$-uniform hypergraph $K_n^r$ yields a monochromatic copy of $H$. We investigate $r_k(H)$ when $k$ grows and $H$ is fixed. For nontrivial 3-uniform hypergraphs $H$, the function $r_k(H)$ ranges from $\sqrt{6k}(1+o(1))$ to double exponential in $k$. We observe that $r_k(H)$ is polynomial in $k$ when $H$ is $r$-partite and at least single-exponential in $k$ otherwise. Erdős, Hajnal and Rado gave bounds for large cliques $K_s^r$ with $s\ge s_0(r)$, showing its correct exponential tower growth. We give a proof for cliques of all sizes, $s>r$, using a slight modification of the celebrated stepping-up lemma of Erdős and Hajnal. For 3-uniform hypergraphs, we give an infinite family with sub-double-exponential upper bound and show connections between graph and hypergraph Ramsey numbers. Specifically, we prove that $$r_k(K_3)\le r_{4k}(K_4^3-e)\le r_{4k}(K_3)+1,$$ where $K_4^3-e$ is obtained from $K_4^3$ by deleting an edge. We provide some other bounds, including single-exponential bounds for $F_5=\{abe,abd,cde\}$ as well as asymptotic or exact values of $r_k(H)$ when $H$ is the bow $\{abc,ade\}$, kite $\{abc,abd\}$, tight path $\{abc,bcd,cde\}$ or the windmill $\{abc,bde,cef,bce\}$. We also determine many new "small" Ramsey numbers and show their relations to designs. For example, the lower bound for $r_6(kite)=8$ is demonstrated by decomposing the triples of $[7]$ into six partial STS (two of them are Fano planes).

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arXiv2013

List Coloring Triangle-Free Hypergraphs

A triangle in a hypergraph is a collection of distinct vertices u,v,w and distinct edges e,f,g with u,v \in e, v,w \in f, w,u \in g, and \{u,v,w\} \cap e \cap f \cap g=\emptyset. The i-degree of a vertex in a hypergraph is the number of edges of size i containing it. We prove that every triangle-free hypergraph of rank three (edges have size two or three) with maximum 3-degree Δ_3 and maximum 2-degree Δ_2 has list chromatic number at most c max{Δ_2/ log{Δ_2}}, (Δ_3 / log{Δ_3})^(1/2)} for some absolute positive constant c. This generalizes a result of Johansson and a result of Frieze and the second author.

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arXiv2012

Coloring the cube with rainbow cycles

For every even positive integer $k\ge 4$ let $f(n,k)$ denote the minimim number of colors required to color the edges of the $n$-dimensional cube $Q_n$, so that the edges of every copy of $k$-cycle $C_k$ receive $k$ distinct colors. Faudree, Gyárfás, Lesniak and Schelp proved that $f(n,4)=n$ for $n=4$ or $n>5$. We consider larger $k$ and prove that if $k \equiv 0$ (mod 4), then there are positive constants $c_1, c_2$ depending only on $k$ such that $$c_1n^{k/4} < f(n,k) < c_2 n^{k/4}.$$ Our upper bound uses an old construction of Bose and Chowla of generalized Sidon sets. For $k \equiv 2$ (mod 4), the situation seems more complicated. For the smallest case k=6 we show that $$n \le f(n, 6) < n^{1+o(1)}.$$ The upper bound is obtained from Behrend's construction of a subset of the integers with no three term arithmetic progression.

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arXiv2012

Specified Intersections

Let M be a subset of {0, .., n} and F be a family of subsets of an n element set such that the size of A intersection B is in M for every A, B in F. Suppose that l is the maximum number of consecutive integers contained in M and n is sufficiently large. Then we prove that |F| < min {1.622^n 100^l, 2^{n/2+l log^2 n}}. The first bound complements the previous bound of roughly (1.99)^n due to Frankl and the second author which applies even when M={0, 1,.., n} - {n/4}. For small l, the second bound above becomes better than the first bound. In this case, it yields 2^{n/2+o(n)} and this can be viewed as a generalization (in an asymptotic sense) of the famous Eventown theorem of Berlekamp. Our second result complements the result of Frankl-Rodl in a different direction. Fix eps>0 and eps n < t < n/5 and let M={0, 1, .., n)-(t, t+n^{0.525}). Then, in the notation above, we prove that for n sufficiently large, |F| < n{n \choose (n+t)/2}. This is essentially sharp aside from the multiplicative factor of n. The short proof uses the Frankl-Wilson theorem and results about the distribution of prime numbers.

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arXiv2011

Counting independent sets in triangle-free graphs

Ajtai, Komlós, and Szemerédi proved that for sufficiently large $t$ every triangle-free graph with $n$ vertices and average degree $t$ has an independent set of size at least $\frac{n}{100t}\log{t}$. We extend this by proving that the number of independent sets in such a graph is at least \[ 2^{(1/2400)\frac{n}{t}\log^2{t}}. \] This result is sharp for infinitely many $t,n$ apart from the constant. An easy consequence of our result is that there exists $c'>0$ such that every $n$-vertex triangle-free graph has at least \[ 2^{c'\sqrt n \log n} \] independent sets. We conjecture that the exponent above can be improved to $\sqrt{n}(\log{n})^{3/2}$. This would be sharp by the celebrated result of Kim which shows that the Ramsey number $R(3,k)$ has order of magnitude $k^2/\log k$.

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