arXiv subjects

Abhishek Methuku

Publications and source records attributed to Abhishek Methuku.

At least 37 records · Page 2Linked to original sources

arXiv2023

Cycles with many chords

How many edges in an $n$-vertex graph will force the existence of a cycle with as many chords as it has vertices? Almost 30 years ago, Chen, Erdős and Staton considered this question and showed that any $n$-vertex graph with $2n^{3/2}$ edges contains such a cycle. We significantly improve this old bound by showing that $Ω(n\log^8n)$ edges are enough to guarantee the existence of such a cycle. Our proof exploits a delicate interplay between certain properties of random walks in almost regular expanders. We argue that while the probability that a random walk of certain length in an almost regular expander is self-avoiding is very small, one can still guarantee that it spans many edges (and that it can be closed into a cycle) with large enough probability to ensure that these two events happen simultaneously.

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arXiv2023

Thresholds for Latin squares and Steiner triple systems: Bounds within a logarithmic factor

We prove that for $n \in \mathbb N$ and an absolute constant $C$, if $p \geq C\log^2 n / n$ and $L_{i,j} \subseteq [n]$ is a random subset of $[n]$ where each $k\in [n]$ is included in $L_{i,j}$ independently with probability $p$ for each $i, j\in [n]$, then asymptotically almost surely there is an order-$n$ Latin square in which the entry in the $i$th row and $j$th column lies in $L_{i,j}$. The problem of determining the threshold probability for the existence of an order-$n$ Latin square was raised independently by Johansson, by Luria and Simkin, and by Casselgren and H{ä}ggkvist; our result provides an upper bound which is tight up to a factor of $\log n$ and strengthens the bound recently obtained by Sah, Sawhney, and Simkin. We also prove analogous results for Steiner triple systems and $1$-factorizations of complete graphs, and moreover, we show that each of these thresholds is at most the threshold for the existence of a $1$-factorization of a nearly complete regular bipartite graph.

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arXiv2023

A proof of the Erdős-Faber-Lovász conjecture

The Erdős-Faber-Lovász conjecture (posed in 1972) states that the chromatic index of any linear hypergraph on $n$ vertices is at most $n$. In this paper, we prove this conjecture for every large $n$. We also provide stability versions of this result, which confirm a prediction of Kahn.

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arXiv2023

Rainbow Turán number of clique subdivisions

We show that for any integer $t\geq 2$, every properly edge-coloured graph on $n$ vertices with more than $n^{1+o(1)}$ edges contains a rainbow subdivision of $K_t$. Note that this bound on the number of edges is sharp up to the $o(1)$ error term. This is a rainbow analogue of some classical results on clique subdivisions and extends some results on rainbow Turán numbers. Our method relies on the framework introduced by Sudakov and Tomon[2020] which we adapt to find robust expanders in the coloured setting.

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arXiv2022

A proof of the Elliott-Rödl conjecture on hypertrees in Steiner triple systems

Hypertrees are linear hypergraphs where every two vertices are connected by a unique path. Elliott and Rödl conjectured that for any given $μ>0$, there exists $n_0$ such that the following holds. Every $n$-vertex Steiner triple system contains all hypertrees with at most $(1-μ)n$ vertices whenever $n\geq n_0$. We prove this conjecture.

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arXiv2021

Graph and hypergraph colouring via nibble methods: A survey

This paper provides a survey of methods, results, and open problems on graph and hypergraph colourings, with a particular emphasis on semi-random `nibble' methods. We also give a detailed sketch of some aspects of the recent proof of the Erdős-Faber-Lovász conjecture.

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arXiv2021

Turán problems for Edge-ordered graphs

In this paper we initiate a systematic study of the Turán problem for edge-ordered graphs. A simple graph is called $\textit{edge-ordered}$, if its edges are linearly ordered. An isomorphism between edge-ordered graphs must respect the edge-order. A subgraph of an edge-ordered graph is itself an edge-ordered graph with the induced edge-order. We say that an edge-ordered graph $G$ $\textit{avoids}$ another edge-ordered graph $H$, if no subgraph of $G$ is isomorphic to $H$. The $\textit{Turán number}$ of an edge-ordered graph $H$ is the maximum number of edges in an edge-ordered graph on $n$ vertices that avoids $H$. We study this problem in general, and establish an Erdős-Stone-Simonovits-type theorem for edge-ordered graphs -- we discover that the relevant parameter for the Turán number of an edge-ordered graph is its $\textit{order chromatic number}$. We establish several important properties of this parameter. We also study Turán numbers of edge-ordered paths, star forests and the cycle of length four. We make strong connections to Davenport-Schinzel theory, the theory of forbidden submatrices, and show an application in Discrete Geometry.

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arXiv2021

Counting cliques in $1$-planar graphs

The problem of maximising the number of cliques among $n$-vertex graphs from various graph classes has received considerable attention. We investigate this problem for the class of $1$-planar graphs where we determine precisely the maximum total number of cliques as well as the maximum number of cliques of any fixed size. We also precisely characterise the extremal graphs for these problems.

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arXiv2021

On the Turán number of the blow-up of the hexagon

The $r$-blowup of a graph $F$, denoted by $F[r]$, is the graph obtained by replacing the vertices and edges of $F$ with independent sets of size $r$ and copies of $K_{r,r}$, respectively. For bipartite graphs $F$, very little is known about the order of magnitude of the Turán number of $F[r]$. In this paper we prove that $\mathrm{ex}(n,C_6[2])=O(n^{5/3})$ and, more generally, for any positive integer $t$, $\mathrm{ex}(n,θ_{3,t}[2])=O(n^{5/3})$. This is tight when $t$ is sufficiently large.

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arXiv2021

Ramsey numbers of Boolean lattices

The poset Ramsey number $R(Q_m,Q_n)$ is the smallest integer $N$ such that any blue-red coloring of the elements of the Boolean lattice $Q_N$ has a blue induced copy of $Q_m$ or a red induced copy of $Q_n$. The weak poset Ramsey number $R_w(Q_m,Q_n)$ is defined analogously, with weak copies instead of induced copies. It is easy to see that $R(Q_m,Q_n) \ge R_w(Q_m,Q_n)$. Axenovich and Walzer showed that $n+2 \le R(Q_2,Q_n) \le 2n+2$. Recently, Lu and Thompson improved the upper bound to $\frac{5}{3}n+2$. In this paper, we solve this problem asymptotically by showing that $R(Q_2,Q_n)=n+O(n/\log n)$. In the diagonal case, Cox and Stolee proved $R_w(Q_n,Q_n) \ge 2n+1$ using a probabilistic construction. In the induced case, Bohman and Peng showed $R(Q_n,Q_n) \ge 2n+1$ using an explicit construction. Improving these results, we show that $R_w(Q_m,Q_n) \ge n+m+1$ for all $m \ge 2$ and large $n$ by giving an explicit construction; in particular, we prove that $R_w(Q_2,Q_n)=n+3$.

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arXiv2021

Minimizing the number of complete bipartite graphs in a $K_s$-saturated graph

A graph $G$ is $F$-saturated if it contains no copy of $F$ as a subgraph but the addition of any new edge to $G$ creates a copy of $F$. We prove that for $s \geq 3$ and $t \geq 2$, the minimum number of copies of $K_{1,t}$ in a $K_s$-saturated graph is $Θ( n^{t/2})$. More precise results are obtained when $t = 2$ where the problem is related to Moore graphs with diameter 2 and girth 5. We prove that for $s \geq 4$ and $t \geq 3$, the minimum number of copies of $K_{2,t}$ in an $n$-vertex $K_s$-saturated graph is at least $Ω( n^{t/5 + 8/5})$ and at most $O(n^{t/2 + 3/2})$. These results answer a question of Chakraborti and Loh. General estimates on the number of copies of $K_{a,b}$ in a $K_s$-saturated graph are also obtained, but finding an asymptotic formula remains open.

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arXiv2020

On $3$-uniform hypergraphs avoiding a cycle of length four

In this note we show that the maximum number of edges in a $3$-uniform hypergraph without a Berge cycle of length four is at most $(1+o(1))\frac{n^{3/2}}{\sqrt{10}}$. This improves earlier estimates by Győri and Lemons and by Füredi and Özkahya.

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arXiv2020

Rainbow Ramsey problems for the Boolean lattice

We address the following rainbow Ramsey problem: For posets $P,Q$ what is the smallest number $n$ such that any coloring of the elements of the Boolean lattice $B_n$ either admits a monochromatic copy of $P$ or a rainbow copy of $Q$. We consider both weak and strong (non-induced and induced) versions of this problem. We also investigate related problems on (partial) $k$-colorings of $B_n$ that do not admit rainbow antichains of size $k$.

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arXiv2020

Combinatorics in the exterior algebra and the Bollobás Two Families Theorem

We investigate the combinatorial structure of subspaces of the exterior algebra of a finite-dimensional real vector space, working in parallel with the extremal combinatorics of hypergraphs. Using initial monomials, projections of the underlying vector space onto subspaces, and the interior product, we find analogues of local and global LYM inequalities, the Erdős-Ko-Rado theorem, and the Ahlswede-Khachatrian bound for $t$-intersecting hypergraphs. Using these tools, we prove a new extension of the Two Families Theorem of Bollobás, giving a weighted bound for subspace configurations satisfying a skew cross-intersection condition. We also verify a recent conjecture of Gerbner, Keszegh, Methuku, Abhishek, Nagy, Patkós, Tompkins, and Xiao on pairs of set systems satisfying both an intersection and a cross-intersection condition.

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arXiv2020

Edge colorings of graphs without monochromatic stars

In this note, we improve on results of Hoppen, Kohayakawa and Lefmann about the maximum number of edge colorings without monochromatic copies of a star of a fixed size that a graph on $n$ vertices may admit. Our results rely on an improved application of an entropy inequality of Shearer.

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arXiv2020

The exact linear Turán number of the Sail

A hypergraph is linear if any two of its edges intersect in at most one vertex. The Sail (or $3$-fan) $F^3$ is the $3$-uniform linear hypergraph consisting of $3$ edges $f_1, f_2, f_3$ pairwise intersecting in the same vertex $v$ and an additional edge $g$ intersecting all $f_i$ in a vertex different from $v$. The linear Turán number $ex_{lin}(n, F^3)$ is the maximum number of edges in a $3$-uniform linear hypergraph on $n$ vertices that does not contain a copy of $F^3$. Füredi and Gyárfás proved that if $n = 3k$, then $ex_{lin}(n, F^3) = k^2$ and the only extremal hypergraphs in this case are transversal designs. They also showed that if $n = 3k+2$, then $ex_{lin}(n, F^3) = k^2+k$, and the only extremal hypergraphs are truncated designs (which are obtained from a transversal design on $3k+3$ vertices with $3$ groups by removing one vertex and all the hyperedges containing it) along with three other small hypergraphs. However, the case when $n =3k+1$ was left open. In this paper, we solve this remaining case by proving that $ex_{lin}(n, F^3) = k^2+1$ if $n = 3k+1$, answering a question of Füredi and Gyárfás. We also characterize all the extremal hypergraphs. The difficulty of this case is due to the fact that these extremal examples are rather non-standard. In particular, they are not derived from transversal designs like in the other cases.

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arXiv2020

Adaptive Majority Problems for Restricted Query Graphs and for Weighted Sets

Suppose that the vertices of a graph $G$ are colored with two colors in an unknown way. The color that occurs on more than half of the vertices is called the majority color (if it exists), and any vertex of this color is called a majority vertex. We study the problem of finding a majority vertex (or show that none exists) if we can query edges to learn whether their endpoints have the same or different colors. Denote the least number of queries needed in the worst case by $m(G)$. It was shown by Saks and Werman that $m(K_n)=n-b(n)$, where $b(n)$ is the number of 1's in the binary representation of $n$. In this paper, we initiate the study of the problem for general graphs. The obvious bounds for a connected graph $G$ on $n$ vertices are $n-b(n)\le m(G)\le n-1$. We show that for any tree $T$ on an even number of vertices we have $m(T)=n-1$ and that for any tree $T$ on an odd number of vertices, we have $n-65\le m(T)\le n-2$. Our proof uses results about the weighted version of the problem for $K_n$, which may be of independent interest. We also exhibit a sequence $G_n$ of graphs with $m(G_n)=n-b(n)$ such that $G_n$ has $O(nb(n))$ edges and $n$ vertices.

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arXiv2019

Generalized rainbow Turán problems

Alon and Shikhelman initiated the systematic study of the following generalized Turán problem: for fixed graphs $H$ and $F$ and an integer $n$, what is the maximum number of copies of $H$ in an $n$-vertex $F$-free graph? An edge-colored graph is called rainbow if all its edges have different colors. The rainbow Turán number of $F$ is defined as the maximum number of edges in a properly edge-colored graph on $n$ vertices with no rainbow copy of $F$. The study of rainbow Turán problems was initiated by Keevash, Mubayi, Sudakov and Verstraëte. Motivated by the above problems, we study the following problem: What is the maximum number of copies of $F$ in a properly edge-colored graph on $n$ vertices without a rainbow copy of $F$? We establish several results, including when $F$ is a path, cycle or tree.

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