arXiv subjects

Abhishek Methuku

Publications and source records attributed to Abhishek Methuku.

At least 55 records · Page 3Linked to original sources

arXiv2019

Bipartite Turán problems for ordered graphs

A zero-one matrix $M$ contains a zero-one matrix $A$ if one can delete some rows and columns of $M$, and turn some 1-entries into 0-entries such that the resulting matrix is $A$. The extremal number of $A$, denoted by $ex(n,A)$, is the maximum number of $1$-entries in an $n\times n$ sized matrix $M$ that does not contain $A$. A matrix $A$ is column-$t$-partite (or row-$t$-partite), if it can be cut along the columns (or rows) into $t$ submatrices such that every row (or column) of these submatrices contains at most one $1$-entry. We prove that if $A$ is column-$t$-partite, then $ex(n,A)<n^{2-\frac{1}{t}+\frac{1}{2t^{2}}+o(1)}$, and if $A$ is both column- and row-$t$-partite, then $ex(n,A)<n^{2-\frac{1}{t}+o(1)}$. Our proof combines a novel density-increment-type argument with the celebrated dependent random choice method. Results about the extremal numbers of zero-one matrices translate into results about the Turán numbers of bipartite ordered graphs. In particular, a zero-one matrix with at most $t$ 1-entries in each row corresponds to a bipartite ordered graph with maximum degree $t$ in one of its vertex classes. Our results are partially motivated by a well known result of Füredi (1991) and Alon, Krivelevich, Sudakov (2003) stating that if $H$ is a bipartite graph with maximum degree $t$ in one of the vertex classes, then $ex(n,H)=O(n^{2-\frac{1}{t}})$. The aim of the present paper is to establish similar general results about the extremal numbers of ordered graphs.

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arXiv2019

Ramsey problems for Berge hypergraphs

For a graph $G$, a hypergraph $\mathcal{H}$ is a Berge copy of $G$ (or a Berge-$G$ in short), if there is a bijection $f : E(G) \rightarrow E(\mathcal{H})$ such that for each $e \in E(G)$ we have $e \subseteq f(e)$. We denote the family of $r$-uniform hypergraphs that are Berge copies of $G$ by $B^rG$. For families of $r$-uniform hypergraphs $\mathbf{H}$ and $\mathbf{H}'$, we denote by $R(\mathbf{H},\mathbf{H}')$ the smallest number $n$ such that in any blue-red coloring of $\mathcal{K}_n^r$ (the complete $r$-uniform hypergraph on $n$ vertices) there is a monochromatic blue copy of a hypergraph in $\mathbf{H}$ or a monochromatic red copy of a hypergraph in $\mathbf{H}'$. $R^c(\mathbf{H})$ denotes the smallest number $n$ such that in any coloring of the hyperedges of $\mathcal{K}_n^r$ with $c$ colors, there is a monochromatic copy of a hypergraph in $\mathbf{H}$. In this paper we initiate the general study of the Ramsey problem for Berge hypergraphs, and show that if $r> 2c$, then $R^c(B^rK_n)=n$. In the case $r = 2c$, we show that $R^c(B^rK_n)=n+1$, and if $G$ is a non-complete graph on $n$ vertices, then $R^c(B^rG)=n$, assuming $n$ is large enough. In the case $r < 2c$ we also obtain bounds on $R^c(B^rK_n)$. Moreover, we also determine the exact value of $R(B^3T_1,B^3T_2)$ for every pair of trees $T_1$ and $T_2$.

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arXiv2019

On the weight of Berge-$F$-free hypergraphs

For a graph $F$, we say a hypergraph is a Berge-$F$ if it can be obtained from $F$ by replacing each edge of $F$ with a hyperedge containing it. A hypergraph is Berge-$F$-free if it does not contain a subhypergraph that is a Berge-$F$. The weight of a non-uniform hypergraph $\mathcal{H}$ is the quantity $\sum_{h \in E(\mathcal{H})} |h|$. Suppose $\mathcal{H}$ is a Berge-$F$-free hypergraph on $n$ vertices. In this short note, we prove that as long as every edge of $\mathcal{H}$ has size at least the Ramsey number of $F$ and at most $o(n)$, the weight of $\mathcal{H}$ is $o(n^2)$. This result is best possible in some sense. Along the way, we study other weight functions, and strengthen results of Gerbner and Palmer; and Grósz, Methuku and Tompkins.

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arXiv2019

Stability results on vertex Turán problems in Kneser graphs

The vertex set of the Kneser graph $K(n,k)$ is $V = \binom{[n]}{k}$ and two vertices are adjacent if the corresponding sets are disjoint. For any graph $F$, the largest size of a vertex set $U \subseteq V$ such that $K(n,k)[U]$ is $F$-free, was recently determined by Alishahi and Taherkhani, whenever $n$ is large enough compared to $k$ and $F$. In this paper, we determine the second largest size of a vertex set $W \subseteq V$ such that $K(n,k)[W]$ is $F$-free, in the case when $F$ is an even cycle or a complete multi-partite graph. In the latter case, we actually give a more general theorem depending on the chromatic number of $F$. These results generalize the celebrated Erd\H os-Ko-Rado theorem and Hilton-Milner theorem.

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arXiv2019

New bounds for a hypergraph Bipartite Turán problem

Let $t$ be an integer such that $t\geq 2$. Let $K_{2,t}^{(3)}$ denote the triple system consisting of the $2t$ triples $\{a,x_i,y_i\}$, $\{b,x_i,y_i\}$ for $1 \le i \le t$, where the elements $a, b, x_1, x_2, \ldots, x_t,$ $y_1, y_2, \ldots, y_t$ are all distinct. Let $ex(n,K_{2,t}^{(3)})$ denote the maximum size of a triple system on $n$ elements that does not contain $K_{2,t}^{(3)}$. This function was studied by Mubayi and Verstraëte, where the special case $t=2$ was a problem of Erdős that was studied by various authors. Mubayi and Verstraëte proved that $ex(n,K_{2,t}^{(3)})<t^4\binom{n}{2}$ and that for infinitely many $n$, $ex(n,K_{2,t}^{(3)})\geq \frac{2t-1}{3} \binom{n}{2}$. These bounds together with a standard argument show that $g(t):=\lim_{n\to \infty} ex(n,K_{2,t}^{(3)})/\binom{n}{2}$ exists and that \[\frac{2t-1}{3}\leq g(t)\leq t^4.\] Addressing the question of Mubayi and Verstraëte on the growth rate of $g(t)$, we prove that as $t \to \infty$, \[g(t) = Θ(t^{1+o(1)}).\]

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arXiv2019

$3$-uniform hypergraphs without a cycle of length five

In this paper we show that the maximum number of hyperedges in a $3$-uniform hypergraph on $n$ vertices without a (Berge) cycle of length five is less than $(0.254 + o(1))n^{3/2}$, improving an estimate of Bollobás and Győri. We obtain this result by showing that not many $3$-paths can start from certain subgraphs of the shadow.

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arXiv2018

Generalized Turán problems for even cycles

Given a graph $H$ and a set of graphs $\mathcal F$, let $ex(n,H,\mathcal F)$ denote the maximum possible number of copies of $H$ in an $\mathcal F$-free graph on $n$ vertices. We investigate the function $ex(n,H,\mathcal F)$, when $H$ and members of $\mathcal F$ are cycles. Let $C_k$ denote the cycle of length $k$ and let $\mathscr C_k=\{C_3,C_4,\ldots,C_k\}$. Some of our main results are the following. (i) We show that $ex(n, C_{2l}, C_{2k}) = Θ(n^l)$ for any $l, k \ge 2$. Moreover, we determine it asymptotically in the following cases: We show that $ex(n,C_4,C_{2k}) = (1+o(1)) \frac{(k-1)(k-2)}{4} n^2$ and that the maximum possible number of $C_6$'s in a $C_8$-free bipartite graph is $n^3 + O(n^{5/2})$. (ii) Solymosi and Wong proved that if Erdős's Girth Conjecture holds, then for any $l \ge 3$ we have $ex(n,C_{2l},\mathscr C_{2l-1})=Θ(n^{2l/(l-1)})$. We prove that forbidding any other even cycle decreases the number of $C_{2l}$'s significantly: For any $k > l$, we have $ex(n,C_{2l},\mathscr C_{2l-1} \cup \{C_{2k}\})=Θ(n^2).$ More generally, we show that for any $k > l$ and $m \ge 2$ such that $2k \neq ml$, we have $ex(n,C_{ml},\mathscr C_{2l-1} \cup \{C_{2k}\})=Θ(n^m).$ (iii) We prove $ex(n,C_{2l+1},\mathscr C_{2l})=Θ(n^{2+1/l}),$ provided a strong version of Erdős's Girth Conjecture holds (which is known to be true when $l = 2, 3, 5$). Moreover, forbidding one more cycle decreases the number of $C_{2l+1}$'s significantly: More precisely, we have $ex(n, C_{2l+1}, \mathscr C_{2l} \cup \{C_{2k}\}) = O(n^{2-\frac{1}{l+1}}),$ and $ex(n, C_{2l+1}, \mathscr C_{2l} \cup \{C_{2k+1}\}) = O(n^2)$ for $l > k \ge 2$. (iv) We also study the maximum number of paths of given length in a $C_k$-free graph, and prove asymptotically sharp bounds in some cases.

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arXiv2018

Triangles in $C_5$-free graphs and Hypergraphs of Girth Six

We introduce a new approach and prove that the maximum number of triangles in a $C_5$-free graph on $n$ vertices is at most $$(1 + o(1)) \frac{1}{3 \sqrt 2} n^{3/2}.$$ We also show a connection to $r$-uniform hypergraphs without (Berge) cycles of length less than six, and estimate their maximum possible size.

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arXiv2018

Few $T$ copies in $H$-saturated graphs

A graph is $F$-saturated if it is $F$-free but the addition of any edge creates a copy of $F$. In this paper we study the quantity $\mathrm{sat}(n, H, F)$ which denotes the minimum number of copies of $H$ that an $F$-saturated graph on $n$ vertices may contain. This parameter is a natural saturation analogue of Alon and Shikhelman's generalized Turán problem, and letting $H = K_2$ recovers the well-studied saturation function. We provide a first investigation into this general function focusing on the cases where the host graph is either $K_s$ or $C_k$-saturated. Some representative interesting behavior is: (a) For any natural number $m$, there are graphs $H$ and $F$ such that $\mathrm{sat}(n, H, F) = Θ(n^m)$. (b) For many pairs $k$ and $l$, we show $\mathrm{sat}(n, C_l, C_k) = 0$. In particular, we prove that there exists a triangle-free $C_k$-saturated graph on $n$ vertices for any $k > 4$ and large enough $n$. (c) $\mathrm{sat}(n, K_3, K_4) = n-2$, $\mathrm{sat}(n, C_4, K_4) \sim \frac{n^2}{2}$, and $\mathrm{sat}(n, C_6, K_5) \sim n^3$. We discuss several intriguing problems which remain unsolved.

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arXiv2018

Asymptotics for Turán numbers of cycles in 3-uniform linear hypergraphs

Let $\mathcal{F}$ be a family of $3$-uniform linear hypergraphs. The linear Turán number of $\mathcal F$ is the maximum possible number of edges in a $3$-uniform linear hypergraph on $n$ vertices which contains no member of $\mathcal{F}$ as a subhypergraph. In this paper we show that the linear Turán number of the five cycle $C_5$ (in the Berge sense) is $\frac{1}{3 \sqrt3}n^{3/2}$ asymptotically. We also show that the linear Turán number of the four cycle $C_4$ and $\{C_3, C_4\}$ are equal asmptotically, which is a strengthening of a theorem of Lazebnik and Verstraëte. We establish a connection between the linear Turán number of the linear cycle of length $2k+1$ and the extremal number of edges in a graph of girth more than $2k-2$. Combining our result and a theorem of Collier-Cartaino, Graber and Jiang, we obtain that the linear Turán number of the linear cycle of length $2k+1$ is $Θ(n^{1+\frac{1}{k}})$ for $k = 2, 3, 4, 6$.

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arXiv2018

On Clique Coverings of Complete Multipartite Graphs

A clique covering of a graph $G$ is a set of cliques of $G$ such that any edge of $G$ is contained in one of these cliques, and the weight of a clique covering is the sum of the sizes of the cliques in it. The sigma clique cover number $scc(G)$ of a graph $G$, is defined as the smallest possible weight of a clique covering of $G$. Let $ K_t(d) $ denote the complete $ t $-partite graph with each part of size $d$. We prove that for any fixed $d \ge 2$, we have $$\lim_{t \rightarrow \infty} scc(K_t(d))= \frac{d}{2} t\log t.$$ This disproves a conjecture of Davoodi, Javadi and Omoomi.

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arXiv2018

General lemmas for Berge-Turán hypergraph problems

For a graph $F$, a hypergraph $\mathcal{H}$ is a Berge copy of $F$ (or a Berge-$F$ in short), if there is a bijection $f : E(F) \rightarrow E(\mathcal{H})$ such that for each $e \in E(F)$ we have $e \subset f(e)$. A hypergraph is Berge-$F$-free if it does not contain a Berge copy of $F$. We denote the maximum number of hyperedges in an $n$-vertex $r$-uniform Berge-$F$-free hypergraph by $\mathrm{ex}_r(n,\textrm{Berge-}F).$ In this paper we prove two general lemmas concerning the maximum size of a Berge-$F$-free hypergraph and use them to establish new results and improve several old results. In particular, we give bounds on $\mathrm{ex}_r(n,\textrm{Berge-}F)$ when $F$ is a path (reproving a result of Győri, Katona and Lemons), a cycle (extending a result of Füredi and Özkahya), a theta graph (improving a result of He and Tait), or a $K_{2,t}$ (extending a result of Gerbner, Methuku and Vizer). We also establish new bounds when $F$ is a clique (which implies extensions of results by Maherani and Shahsiah and by Gyárfás) and when $F$ is a general tree.

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arXiv2018

Avoiding long Berge cycles, the missing cases $k=r+1$ and $k = r+2$

The maximum size of an $r$-uniform hypergraph without a Berge cycle of length at least $k$ has been determined for all $k \ge r+3$ by Füredi, Kostochka and Luo and for $k<r$ (and $k=r$, asymptotically) by Kostochka and Luo. In this paper, we settle the remaining cases: $k=r+1$ and $k=r+2$, proving a conjecture of Füredi, Kostochka and Luo.

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arXiv2018

Asymptotics for the Turán number of Berge-$K_{2,t}$

Let $F$ be a graph. A hypergraph is called Berge-$F$ if it can be obtained by replacing each edge of $F$ by a hyperedge containing it. Let $\mathcal{F}$ be a family of graphs. The Turán number of Berge-$\mathcal{F}$ is the maximum possible number of edges in an $r$-uniform hypergraph on $n$ vertices containing no Berge-$F$ as a subhypergraph (for every $F \in \mathcal{F}$) and is denoted by $ex_r(n,\mathcal{F})$. We determine the asymptotics for the Turán number of Berge-$K_{2,t}$ by showing $$ex_3(n,K_{2,t})=\frac{1}{6}(t-1)^{3/2} \cdot n^{3/2}(1+o(1))$$ for any given $t \ge 7$. We study the analogous question for linear hypergraphs and show that $$ex_3(n,\{C_2, K_{2,t}\}) = \frac{1}{6}\sqrt{t-1} \cdot n^{3/2}(1+o_{t}(1)).$$ We also prove general upper and lower bounds on the Turán numbers of a class of graphs including $ex_r(n, K_{2,t})$, $ex_r(n,\{C_2, K_{2,t}\})$, and $ex_r(n, C_{2k})$ for $r \ge 3$. Our bounds improve results of Gerbner and Palmer, Füredi and Özkahya, Timmons, and provide a new proof of a result of Jiang and Ma.

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arXiv2018

Vertex Turán problems for the oriented hypercube

In this short note we consider the oriented vertex Turán problem in the hypercube: for a fixed oriented graph $\overrightarrow{F}$, determine the maximum size $ex_v(\overrightarrow{F}, \overrightarrow{Q_n})$ of a subset $U$ of the vertices of the oriented hypercube $\overrightarrow{Q_n}$ such that the induced subgraph $\overrightarrow{Q_n}[U]$ does not contain any copy of $\overrightarrow{F}$. We obtain the exact value of $ex_v(\overrightarrow{P_k}, \overrightarrow{Q_n})$ for the directed path $\overrightarrow{P_k}$, the exact value of $ex_v(\overrightarrow{V_2}, \overrightarrow{Q_n})$ for the directed cherry $\overrightarrow{V_2}$ and the asymptotic value of $ex_v(\overrightarrow{T}, \overrightarrow{Q_n})$ for any directed tree $\overrightarrow{T}$.

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arXiv2018

Linearity of Saturation for Berge Hypergraphs

For a graph $F$, we say a hypergraph $H$ is Berge-$F$ if it can be obtained from $F$ be replacing each edge of $F$ with a hyperedge containing it. We say a hypergraph is Berge-$F$-saturated if it does not contain a Berge-$F$, but adding any hyperedge creates a copy of Berge-$F$. The $k$-uniform saturation number of Berge-$F$, $\mathrm{sat}_k(n,\text{Berge-}F)$ is the fewest number of edges in a Berge-$F$-saturated $k$-uniform hypergraph on $n$ vertices. We show that $\mathrm{sat}_k(n,\text{Berge-}F) = O(n)$ for all graphs $F$ and uniformities $3\leq k\leq 5$, partially answering a conjecture of English, Gordon, Graber, Methuku, and Sullivan. We also extend this conjecture to Berge copies of hypergraphs.

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arXiv2018

On the Turán number of some ordered even cycles

A classical result of Bondy and Simonovits in extremal graph theory states that if a graph on $n$ vertices contains no cycle of length $2k$ then it has at most $O(n^{1+1/k})$ edges. However, matching lower bounds are only known for $k=2,3,5$. In this paper we study ordered variants of this problem and prove some tight estimates for a certain class of ordered cycles that we call bordered cycles. In particular, we show that the maximum number of edges in an ordered graph avoiding bordered cycles of length at most $2k$ is $Θ(n^{1+1/k})$. Strengthening the result of Bondy and Simonovits in the case of 6-cycles, we also show that it is enough to forbid these bordered orderings of the 6-cycle to guarantee an upper bound of $O(n^{4/3})$ on the number of edges.

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arXiv2018

Generalized Turán problems for disjoint copies of graphs

Given two graphs $H$ and $F$, the maximum possible number of copies of $H$ in an $F$-free graph on $n$ vertices is denoted by $ex(n,H,F)$. We investigate the function $ex(n,H,kF)$, where $kF$ denotes $k$ vertex disjoint copies of a fixed graph $F$. Our results include cases when $F$ is a complete graph, cycle or a complete bipartite graph.

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