arXiv subjects

Dhruv Mubayi

Publications and source records attributed to Dhruv Mubayi.

At least 37 records · Page 2Linked to original sources

arXiv2024

Combining the theorems of Turán and de Bruijn-Erd\H os

Fix an integer $s \ge 2$. Let $\mathcal{P}$ be a set of $n$ points and let $\mathcal{L}$ be a set of lines in a linear space such that no line in $\mathcal{L}$ contains more than $(n-1)/(s-1)$ points of $\mathcal{P}$. Suppose that for every $s$-set $S$ in $\mathcal{P}$, there is a pair of points in $S$ that lies in a line from $\mathcal{L}$. We prove that $|\mathcal{L}| \ge (n-1)/(s-1)+s-1$ for $n$ large, and this is sharp when $n-1$ is a multiple of $s-1$. This generalizes the de Bruijn-Erd\H os theorem which is the case $s=2$. Our result is proved in the more general setting of linear hypergraphs.

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arXiv2024

Off-diagonal Ramsey numbers for slowly growing hypergraphs

For a $k$-uniform hypergraph $F$ and a positive integer $n$, the Ramsey number $r(F,n)$ denotes the minimum $N$ such that every $N$-vertex $F$-free $k$-uniform hypergraph contains an independent set of $n$ vertices. A hypergraph is $\textit{slowly growing}$ if there is an ordering $e_1,e_2,\dots,e_t$ of its edges such that $|e_i \setminus \bigcup_{j = 1}^{i - 1}e_j| \leq 1$ for each $i \in \{2, \ldots, t\}$. We prove that if $k \geq 3$ is fixed and $F$ is any non $k$-partite slowly growing $k$-uniform hypergraph, then for $n\ge2$, \[ r(F,n) = Ω\Bigl(\frac{n^k}{(\log n)^{2k - 2}}\Bigr).\] In particular, we deduce that the off-diagonal Ramsey number $r(F_5,n)$ is of order $n^{3}/\mbox{polylog}(n)$, where $F_5$ is the triple system $\{123, 124, 345\}$. This is the only 3-uniform Berge triangle for which the polynomial power of its off-diagonal Ramsey number was not previously known. Our constructions use pseudorandom graphs, martingales, and hypergraph containers.

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arXiv2024

Erdős-Rogers functions for arbitrary pairs of graphs

Let $f_{F,G}(n)$ be the largest size of an induced $F$-free subgraph that every $n$-vertex $G$-free graph is guaranteed to contain. We prove that for any triangle-free graph $F$, \[ f_{F,K_3}(n) = f_{K_2,K_3}(n)^{1 + o(1)} = n^{\frac{1}{2} + o(1)}.\] Along the way we give a slight improvement of a construction of Erd\H os-Frankl-Rödl for the Brown-Erd\H os-Sós $(3r-3,3)$-problem when $r$ is large. In contrast to our result for $K_3$, for any $K_4$-free graph $F$ containing a cycle, we prove there exists $c_F > 0$ such that $$f_{F,K_4}(n) > f_{K_2,K_4}(n)^{1 + c_F} = n^{\frac{1}{3}+c_F+o(1)}.$$ \iffalse We also observe that our earlier proof for $F=K_3$ generalizes to $f_{F,K_4}(n) = O(\sqrt{n}\log n)$ for all $F$ containing a cycle. \fi For every graph $G$, we prove that there exists $\varepsilon_G >0$ such that whenever $F$ is a non-empty graph such that $G$ is not contained in any blowup of $F$, then $f_{F,G}(n) = O(n^{1-\varepsilon_G})$. On the other hand, for graph $G$ that is not a clique, and every $\varepsilon>0$, we exhibit a $G$-free graph $F$ such that $f_{F,G}(n) = Ω(n^{1-\varepsilon})$.

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arXiv2024

Big line or big convex polygon

Let $ES_{\ell}(n)$ be the minimum $N$ such that every $N$-element point set in the plane contains either $\ell$ collinear members or $n$ points in convex position. We prove that there is a constant $C>0$ such that, for each $\ell, n \ge 3$, $$ (3\ell - 1) \cdot 2^{n-5} < ES_{\ell}(n) < \ell^2 \cdot 2^{n+ C\sqrt{n\log n}}.$$ A similar extension of the well-known Erd\H os--Szekeres cups-caps theorem is also proved.

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arXiv2024

Inducibility of rainbow graphs

Fix $k\ge 11$ and a rainbow $k$-clique $R$. We prove that the inducibility of $R$ is $k!/(k^k-k)$. An extremal construction is a balanced recursive blow-up of $R$. This answers a question posed by Huang, that is a generalization of an old problem of Erd\H os and Sós. It remains open to determine the minimum $k$ for which our result is true. More generally, we prove that there is an absolute constant $C>0$ such that every $k$-vertex connected rainbow graph with minimum degree at least $C\log k$ has inducibility $k!/(k^k-k)$.

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arXiv2024

Ramsey numbers and the Zarankiewicz problem

Building on recent work of Mattheus and Verstraëte, we establish a general connection between Ramsey numbers of the form $r(F,t)$ for $F$ a fixed graph and a variant of the Zarankiewicz problem asking for the maximum number of 1s in an $m$ by $n$ $0/1$-matrix that does not have any matrix from a fixed finite family $\mathcal{L}(F)$ derived from $F$ as a submatrix. As an application, we give new lower bounds for the Ramsey numbers $r(C_5,t)$ and $r(C_7,t)$, namely, $r(C_5,t) = \tildeΩ(t^{\frac{10}{7}})$ and $r(C_7,t) = \tildeΩ(t^{\frac{5}{4}})$. We also show how the truth of a plausible conjecture about Zarankiewicz numbers would allow an approximate determination of $r(C_{2\ell+1}, t)$ for any fixed integer $\ell \geq 2$.

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arXiv2024

On off-diagonal hypergraph Ramsey numbers

A fundamental problem in Ramsey theory is to determine the growth rate in terms of $n$ of the Ramsey number $r(H, K_n^{(3)})$ of a fixed $3$-uniform hypergraph $H$ versus the complete $3$-uniform hypergraph with $n$ vertices. We study this problem, proving two main results. First, we show that for a broad class of $H$, including links of odd cycles and tight cycles of length not divisible by three, $r(H, K_n^{(3)}) \ge 2^{Ω_H(n \log n)}$. This significantly generalizes and simplifies an earlier construction of Fox and He which handled the case of links of odd cycles and is sharp both in this case and for all but finitely many tight cycles of length not divisible by three. Second, disproving a folklore conjecture in the area, we show that there exists a linear hypergraph $H$ for which $r(H, K_n^{(3)})$ is superpolynomial in $n$. This provides the first example of a separation between $r(H,K_n^{(3)})$ and $r(H,K_{n,n,n}^{(3)})$, since the latter is known to be polynomial in $n$ when $H$ is linear.

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arXiv2024

Ordered and colored subgraph density problems

We consider three extremal problems about the number of copies of a fixed graph in another larger graph. First, we correct an error in a result of Reiher and Wagner and prove that the number of $k$-edge stars in a graph with density $x \in [0, 1]$ is asymptotically maximized by a clique and isolated vertices or its complement. Next, among ordered $n$-vertex graphs with $m$ edges, we determine the maximum and minimum number of copies of a $k$-edge star whose nonleaf vertex is minimum among all vertices of the star. Finally, for $s \ge 2$, we define a particular $3$-edge-colored complete graph $F$ on $2s$ vertices with colors blue, green and red, and determine, for each $(x_b, x_g)$ with $x_b+x_g\le 1$ and $x_b, x_g \ge 0$, the maximum density of $F$ in a large graph whose blue, green and red edge sets have densities $x_b, x_g$ and $1-x_b-x_g$, respectively. These are the first nontrivial examples of colored graphs for which such complete results are proved.

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arXiv2024

On the order of Erdős-Rogers functions

For an integer $n \geq 1$, the Erdős-Rogers function $f_{s}(n)$ is the maximum integer $m$ such that every $n$-vertex $K_{s+1}$-free graph has a $K_s$-free subgraph with $m$ vertices. It is known that for all $s \geq 3$, $f_{s}(n) = Ω(\sqrt{n\log n}/\log \log n)$ as $n \rightarrow \infty$. In this paper, we show that for all $s \geq 3$, \begin{equation*} f_{s}(n) = O(\sqrt{n}\, \log n). \end{equation*} This improves previous bounds of order $\sqrt{n} (\log n)^{2(s + 1)^2}$ by Dudek, Retter and Rödl.

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arXiv2023

Hypergraphs with infinitely many extremal constructions

We give the first exact and stability results for a hypergraph Turán problem with infinitely many extremal constructions that are far from each other in edit-distance. This includes an example of triple systems with Turán density $2/9$, thus answering some questions posed by the third and fourth authors and Reiher about the feasible region of hypergraphs. Our results also provide extremal constructions whose shadow density is a transcendental number. Our novel approach is to construct certain multilinear polynomials that attain their maximum (in the standard simplex) on a line segment and then to use these polynomials to define an operation on hypergraphs that gives extremal constructions.

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arXiv2023

Large cliques or co-cliques in hypergraphs with forbidden order-size pairs

The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $ε>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^ε$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every $S \subseteq \{0,1,2,3,4\}$, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. In most cases the bounds are essentially tight. We also determine, for all $S$, whether the growth rate is polynomial or polylogarithmic. Some open problems remain.

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arXiv2023

On The Random Turán number of linear cycles

Given two $r$-uniform hypergraphs $G$ and $H$ the Turán number $\rm{ex}(G, H)$ is the maximum number of edges in an $H$-free subgraph of $G$. We study the typical value of $\rm{ex}(G, H)$ when $G=G_{n,p}^{(r)}$, the Erdős-Rényi random $r$-uniform hypergraph, and $H=C_{2\ell}^{(r)}$, the $r$-uniform linear cycle of length $2\ell$. The case of graphs ($r=2$) is a longstanding open problem that has been investigated by many researchers. We determine the order of magnitude of $\rm{ex}\left(G_{n,p}^{(r)}, C_{2\ell}^{(r)}\right)$ for all $r\geq 4$ and all $\ell\geq 2$ up to polylogarithmic factors for all values of $p=p(n)$. Our proof is based on the container method and uses a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan.

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arXiv2023

Coloring hypergraphs that are the union of nearly disjoint cliques

We consider the maximum chromatic number of hypergraphs consisting of cliques that have pairwise small intersections. Designs of the appropriate parameters produce optimal constructions, but these are known to exist only when the number of cliques is exponential in the clique size. We construct near designs where the number of cliques is polynomial in the clique size, and show that they have large chromatic number. The case when the cliques have pairwise intersections of size at most one seems particularly challenging. Here we give lower bounds by analyzing a random greedy hypergraph process. We also consider the related question of determining the maximum number of caps in a finite projective/affine plane and obtain nontrivial upper and lower bounds.

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arXiv2023

Ramsey numbers of cliques versus monotone paths

One formulation of the Erdos-Szekeres monotone subsequence theorem states that for any red/blue coloring of the edge set of the complete graph on $\{1, 2, \ldots, N\}$, there exists a monochromatic red $s$-clique or a monochromatic blue increasing path $P_n$ with $n$ vertices, provided $N >(s-1)(n-1)$. %We had previously shown that a suitable generalization of this problem to quadruple systems is essentially equivalent to classical diagonal hypergraph Ramsey numbers. Here, we prove a similar statement as above in the off-diagonal case for triple systems, with the quasipolynomial bound $N>2^{c(\log n)^{s-1}}$. For the $t$th power $P_n^t$ of the ordered increasing graph path with $n$ vertices, we prove a near linear bound $c\, n(\log n)^{s-2}$ which improves the previous bound that applied to a more general class of graphs than $P_n^t$ due to Conlon-Fox-Lee-Sudakov.

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arXiv2023

Triangles in graphs without bipartite suspensions

Given graphs $T$ and $H$, the generalized Turán number ex$(n,T,H)$ is the maximum number of copies of $T$ in an $n$-vertex graph with no copies of $H$. Alon and Shikhelman, using a result of Erd\H os, determined the asymptotics of ex$(n,K_3,H)$ when the chromatic number of $H$ is greater than 3 and proved several results when $H$ is bipartite. We consider this problem when $H$ has chromatic number 3. Even this special case for the following relatively simple 3-chromatic graphs appears to be challenging. The suspension $\widehat H$ of a graph $H$ is the graph obtained from $H$ by adding a new vertex adjacent to all vertices of $H$. We give new upper and lower bounds on ex$(n,K_3,\widehat{H})$ when $H$ is a path, even cycle, or complete bipartite graph. One of the main tools we use is the triangle removal lemma, but it is unclear if much stronger statements can be proved without using the removal lemma.

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arXiv2023

Homogeneous sets in hypergraphs with forbidden order-size pairs

The well-known Erdős-Hajnal conjecture states that for any graph $F$, there exists $ε>0$ such that every $n$-vertex graph $G$ that contains no induced copy of $F$ has a homogeneous set of size at least $n^ε$. We consider a variant of the Erdős-Hajnal problem for hypergraphs where we forbid a family of hypergraphs described by their orders and sizes. For graphs, we observe that if we forbid induced subgraphs on $m$ vertices and $f$ edges for any positive $m$ and $0\leq f \leq \binom{m}{2}$, then we obtain large homogeneous sets. For triple systems, in the first nontrivial case $m=4$, for every $S \subseteq \{0,1,2,3,4\}$, we give bounds on the minimum size of a homogeneous set in a triple system where the number of edges spanned by every four vertices is not in $S$. For all $S$ we determine if the growth rate is polylogarithmic. Several open problems remain.

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arXiv2023

Turán problems in pseudorandom graphs

Given a graph $F$, we consider the problem of determining the densest possible pseudorandom graph that contains no copy of $F$. We provide an embedding procedure that improves a general result of Conlon, Fox, and Zhao which gives an upper bound on the density. In particular, our result implies that optimally pseudorandom graphs with density greater than $n^{-1/3}$ must contain a copy of the Peterson graph, while the previous best result gives the bound $n^{-1/4}$. Moreover, we conjecture that the exponent $1/3$ in our bound is tight. We also construct the densest known pseudorandom $K_{2,3}$-free graphs that are also triangle-free. Finally, we obtain the densest known construction of clique-free pseudorandom graphs due to Bishnoi, Ihringer and Pepe in a novel way and give a different proof that they have no large clique.

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arXiv2022

Hypergraph Ramsey numbers of cliques versus stars

Let $K_m^{(3)}$ denote the complete $3$-uniform hypergraph on $m$ vertices and $S_n^{(3)}$ the $3$-uniform hypergraph on $n+1$ vertices consisting of all $\binom{n}{2}$ edges incident to a given vertex. Whereas many hypergraph Ramsey numbers grow either at most polynomially or at least exponentially, we show that the off-diagonal Ramsey number $r(K_{4}^{(3)},S_n^{(3)})$ exhibits an unusual intermediate growth rate, namely, \[ 2^{c \log^2 n} \le r(K_{4}^{(3)},S_n^{(3)}) \le 2^{c' n^{2/3}\log n} \] for some positive constants $c$ and $c'$. The proof of these bounds brings in a novel Ramsey problem on grid graphs which may be of independent interest: what is the minimum $N$ such that any $2$-edge-coloring of the Cartesian product $K_N \square K_N$ contains either a red rectangle or a blue $K_n$?

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