arXiv subjects

Maria Axenovich

Publications and source records attributed to Maria Axenovich.

At least 37 records · Page 2Linked to original sources

arXiv2023

A class of graphs of zero Turán density in a hypercube

A graph is cubical if it is a subgraph of a hypercube. For a cubical graph $H$ and a hypercube $Q_n$, $ex(Q_n, H)$ is the largest number of edges in an $H$-free subgraph of $Q_n$. If $ex(Q_n, H)$ is at least a positive proportion of the number of edges in $Q_n$, $H$ is said to have a positive Turán density in a hypercube or simply a positive Turán density; otherwise it has a zero Turán density. Determining $ex(Q_n, H)$ and even identifying whether $H$ has a positive or a zero Turán density remains a widely open question for general $H$. By relating extremal numbers in a hypercube and certain corresponding hypergraphs, Conlon found a large class of cubical graphs, ones having so-called partite representation, that have a zero Turán density. He raised a question whether this gives a characterisation, i.e., whether a cubical graph has zero Turán density if and only if it has partite representation. Here, we show that, as suspected by Conlon, this is not the case. We give an example of a class of cubical graphs which have no partite representation, but on the other hand, have a zero Turán density. In addition, we show that any graph whose every block has partite representation has a zero Turán density in a hypercube.

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arXiv2023

Poset Ramsey number $R(P,Q_n)$. II. N-shaped poset

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f\colon P\rightarrow P'$ and for any $A, B\in P$, $A\leq _P B$ if and only if $f(A)\leq_{P'} f(B)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on the poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. It is known that $n+c_1(P) \leq R(P,Q_n) \leq c_2(P) n$, for positive constants $c_1$ and $c_2$. However, there is no poset $P$ known, for which $R(P, Q_n)> (1+ε)n$, for $ε>0$. This paper is devoted to a new method for finding upper bounds on $R(P, Q_n)$ using a duality between copies of $Q_n$ and sets of elements that cover them, referred to as blockers. We prove several properties of blockers and their direct relation to the Ramsey numbers. Using these properties we show that $R(\mathcal{N},Q_n)=n+Θ(n/\log n)$, for a poset $\mathcal{N}$ with four elements $A, B, C, $ and $D$, such that $A<C$, $B<D$, $B<C$, and the remaining pairs of elements are incomparable.

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arXiv2023

A note on interval colourings of graphs

A graph is said to be interval colourable if it admits a proper edge-colouring using palette $\mathbb{N}$ in which the set of colours incident to each vertex is an interval. The interval colouring thickness of a graph $G$ is the minimum $k$ such that $G$ can be edge-decomposed into $k$ interval colourable graphs. We show that $θ(n)$, the maximum interval colouring thickness of an $n$-vertex graph, satisfies $θ(n) =Ω(\log(n)/\log\log(n))$ and $θ(n)\leq n^{5/6+o(1)}$, which improves on the trivial lower bound and an upper bound of the first author and Zheng. As a corollary, we answer a question of Asratian, Casselgren, and Petrosyan and disprove a conjecture of Borowiecka-Olszewska, Drgas-Burchardt, Javier-Nol, and Zuazua. We also confirm a conjecture of the first author that any interval colouring of an $n$-vertex planar graph uses at most $3n/2-2$ colours.

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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

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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arXiv2022

Rainbow Subgraphs in Edge-colored Complete Graphs -- Answering two Questions by Erdős and Tuza

An edge-coloring of a complete graph with a set of colors $C$ is called completely balanced if any vertex is incident to the same number of edges of each color from $C$. Erdős and Tuza asked in $1993$ whether for any graph $F$ on $\ell$ edges and any completely balanced coloring of any sufficiently large complete graph using $\ell$ colors contains a rainbow copy of $F$. This question was restated by Erdős in his list of ``Some of my favourite problems on cycles and colourings''. We answer this question in the negative for most cliques $F=K_q$ by giving explicit constructions of respective completely balanced colorings. Further, we answer a related question concerning completely balanced colorings of complete graphs with more colors than the number of edges in the graph $F$.

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arXiv2022

Extremal numbers for cycles in a hypercube

Let $ex(Q_n, H)$ be the largest number of edges in a subgraph $G$ of a hypercube $Q_n$ such that there is no subgraph of $G$ isomorphic to $H$. We show that for any integer $k\geq 3$, $$ex(Q_n, C_{4k+2})= O(n^{\frac{5}{6} + \frac{1}{3(2k-2)}} 2^n).$$

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arXiv2022

Unavoidable order-size pairs in hypergraphs -- positive forcing density

Erdős, Füredi, Rothschild and Sós initiated a study of classes of graphs that forbid every induced subgraph on a given number $m$ of vertices and number $f$ of edges. Extending their notation to $r$-graphs, we write $(n,e) \to_r (m,f)$ if every $r$-graph $G$ on $n$ vertices with $e$ edges has an induced subgraph on $m$ vertices and $f$ edges. The \emph{forcing density} of a pair $(m,f)$ is $$ σ_r(m,f) =\left. \limsup\limits_{n \to \infty} \frac{|\{e : (n,e) \to_r (m,f)\}|}{\binom{n}{r}} \right. .$$ In the graph setting it is known that there are infinitely many pairs $(m, f)$ with positive forcing density. Weber asked if there is a pair of positive forcing density for $r\geq 3$ apart from the trivial ones $(m, 0)$ and $(m, \binom{m}{r})$. Answering her question, we show that $(6,10)$ is such a pair for $r=3$ and conjecture that it is the unique such pair. Further, we find necessary conditions for a pair to have positive forcing density, supporting this conjecture.

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arXiv2022

Interval colorings of graphs -- coordinated and unstable no-wait schedules

A proper edge-coloring of a graph is an interval coloring if the labels on the edges incident to any vertex form an interval of consecutive integers. Interval thickness s(G) of a graph G is the smallest number of interval colorable graphs edge-decomposing G. We prove that s(G)=o(n) for any graph G on n vertices. This improves the previously known bound of 2n/5 by Asratian, Casselgren, and Petrosyan. While we do not have a single example of a graph with interval thickness strictly greater than 2, we construct bipartite graphs whose interval spectrum has arbitrarily many arbitrarily large gaps. Here, an interval spectrum of a graph is the set of all integers t such that the graph has an interval coloring using t colors. Interval colorings of bipartite graphs naturally correspond to no-wait schedules, say for parent-teacher conferences, where a conversation between any teacher and any parent lasts the same amount of time. Our results imply that any such conference with $n$ participants can be coordinated in o(n) no-wait periods. In addition, we show that for any integers t and T, t<T, there is a set of pairs of parents and teachers wanting to talk to each other, such that any no-wait schedules are unstable -- they could last t hours and could last T hours, but there is no possible no-wait schedule lasting x hours if t<x<T.

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arXiv2022

Generalized Turán densities in the hypercube

A classical extremal, or Turán-type problem asks to determine ${\rm ex}(G, H)$, the largest number of edges in a subgraph of a graph $G$ which does not contain a subgraph isomorphic to $H$. Alon and Shikhelman introduced the so-called generalized extremal number ${\rm ex}(G,T,H)$, defined to be the maximum number of subgraphs isomorphic to $T$ in a subgraph of $G$ that contains no subgraphs isomorphic to $H$. In this paper we investigate the case when $G = Q_n$, the hypercube of dimension $n$, and $T$ and $H$ are smaller hypercubes or cycles.

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arXiv2021

Poset Ramsey numbers: large Boolean lattice versus a fixed poset

Given partially ordered sets (posets) $(P, \leq_P)$ and $(P', \leq_{P'})$, we say that $P'$ contains a copy of $P$ if for some injective function $f: P\rightarrow P'$ and for any $X, Y\in P$, $X\leq _P Y$ if and only of $f(X)\leq_{P'} f(Y)$. For any posets $P$ and $Q$, the poset Ramsey number $R(P,Q)$ is the least positive integer $N$ such that no matter how the elements of an $N$-dimensional Boolean lattice are colored in blue and red, there is either a copy of $P$ with all blue elements or a copy of $Q$ with all red elements. We focus on a poset Ramsey number $R(P, Q_n)$ for a fixed poset $P$ and an $n$-dimensional Boolean lattice $Q_n$, as $n$ grows large. We show a sharp jump in behaviour of this number as a function of $n$ depending on whether or not $P$ contains a copy of either a poset $V$, i.e. a poset on elements $A, B, C$ such that $B>C$, $A>C$, and $A$ and $B$ incomparable, or a poset $Λ$, its symmetric counterpart. Specifically, we prove that if $P$ contains a copy of $V$ or $Λ$ then $R(P, Q_n) \geq n +\frac{1}{15} \frac{n}{\log n}$. Otherwise $R(P, Q_n) \leq n + c(P)$ for a constant $c(P)$. This gives the first non-marginal improvement of a lower bound on poset Ramsey numbers and as a consequence gives $R(Q_2, Q_n) = n + Θ(\frac{n}{\log n})$.

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arXiv2021

Canonical theorems for colored integers with respect to some linear combinations

Hindman proved in 1979 that no matter how natural numbers are colored in r colors, for a fixed positive integer r, there is an infinite subset X of numbers and a color t such that for any finite non-empty subset X' of X, the color of the sum of elements from X' is t. Later, Taylor extended this result to colorings with unrestricted number of colors and five unavoidable color patterns on finite sums. This result is referred to as a canonization of Hindman's theorem and parallels the Canonical Ramsey Theorem of Erdos and Rado. We extend Taylor's result from sums, that are linear combinations with coefficients 1, to several linear combinations with coefficients 1 and -1. These results in turn could be interpreted as canonical-type theorems for solutions to infinite systems.

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arXiv2021

The Erdős-Hajnal conjecture for three colors and multiple forbidden patterns

Erdős and Szekeres's quantitative version of Ramsey's theorem asserts that any complete graph on n vertices that is edge-colored with two colors has a monochromatic clique on at least 1/2log(n) vertices. The famous Erdős-Hajnal conjecture asserts that forbidding fixed color patterns ensures larger monochromatic cliques. Specifically, it claims that for any fixed integer k and any clique K on k vertices edge-colored with two colors, there is a positive constant a such that in any complete n-vertex graph edge-colored with two colors that does not contain a copy of K, there is a monochromatic clique on at least n^a vertices. We consider edge-colorings with three colors. For a family H of triangles, each colored with colors from {r, b, y}, Forb(n,H) denotes a family of edge-colorings of the complete n-vertex graph using colors from {r, b, y} and containing none of the colorings from H. Let h_2(n, H) be the maximum q such that any coloring from Forb(n, H) has a clique on at least q vertices using at most two colors. We provide bounds on h_2(n, H) for all families H consisting of at most three triangles. For most of them, our bounds are asymptotically tight. This extends a result of Fox, Grinshpun, and Pach, who determined h_2(n, H) for H consisting of a rainbow triangle, and confirms the multicolor Erdős-Hajnal conjecture for these sets of patterns.

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arXiv2021

Absolutely avoidable order-size pairs for induced subgraphs

We call a pair $(m,f)$ of integers, $m\geq 1$, $0\leq f \leq \binom{m}{2}$, \emph{absolutely avoidable} if there is $n_0$ such that for any pair of integers $(n,e)$ with $n>n_0$ and $0\leq e\leq \binom{n}{2}$ there is a graph on $n$ vertices and $e$ edges that contains no induced subgraph on $m$ vertices and $f$ edges. Some pairs are clearly not absolutely avoidable, for example $(m,0)$ is not absolutely avoidable since any sufficiently sparse graph on at least $m$ vertices contains independent sets on $m$ vertices. Here we show that there are infinitely many absolutely avoidable pairs. We give a specific infinite set $M$ such that for any $m\in M$, the pair $(m, \binom{m}{2}/2)$ is absolutely avoidable. In addition, among other results, we show that for any monotone integer function $q(m)$, $|q(m)|=O(m)$, there are infinitely many values of $m$ such that the pair $(m, \binom{m}{2}/2 +q(m))$ is absolutely avoidable.

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arXiv2021

Sum-distinguishing number of sparse hypergraphs

A vertex labeling of a hypergraph is sum distinguishing if it uses positive integers and the sums of labels taken over the distinct hyperedges are distinct. Let s(H) be the smallest integer N such that there is a sum-distinguishing labeling of H with each label at most N. The largest value of s(H) over all hypergraphs on n vertices and m hyperedges is denoted s(n,m). We prove that s(n,m) is almost-quadratic in m as long as m is not too large. More precisely, the following holds: If n < m < n^{O(1)} then s(n,m)= m^2/w(m), where w(m) is a function that goes to infinity and is smaller than any polynomial in m. The parameter s(n,m) has close connections to several other graph and hypergraph functions, such as the irregularity strength of hypergraphs. Our result has several applications, notably: 1. We answer a question of Gyarfas et al. whether there are n-vertex hypergraphs with irregularity strength greater than 2n. In fact we show that there are n-vertex hypergraphs with irregularity strength at least n^{2-o(1)}. 2. Our results imply that s*(n)=n^2/w(n) where s*(n) is the distinguishing closed-neighborhood number, i.e., the smallest integer N such that any n-vertex graph allows for a vertex labeling with positive integers at most N so that the sums of labels on distinct closed neighborhoods of vertices are distinct.

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arXiv2021

Long path and cycle decompositions of even hypercubes

We consider edge decompositions of the $n$-dimensional hypercube $Q_n$ into isomorphic copies of a given graph $H$. While a number of results are known about decomposing $Q_n$ into graphs from various classes, the simplest cases of paths and cycles of a given length are far from being understood. A conjecture of Erde asserts that if $n$ is even, $\ell < 2^n$ and $\ell$ divides the number of edges of $Q_n$, then the path of length $\ell$ decomposes $Q_n$. Tapadia et al.\ proved that any path of length $2^mn$, where $2^m<n$, satisfying these conditions decomposes $Q_n$. Here, we make progress toward resolving Erde's conjecture by showing that cycles of certain lengths up to $2^{n+1}/n$ decompose $Q_n$. As a consequence, we show that $Q_n$ can be decomposed into copies of any path of length at most $2^{n}/n$ dividing the number of edges of $Q_n$, thereby settling Erde's conjecture up to a linear factor.

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arXiv2020

Strong complete minors in digraphs

Kostochka and Thomason independently showed that any graph with average degree $Ω(r\sqrt{\log r})$ contains a $K_r$ minor. In particular, any graph with chromatic number $Ω(r\sqrt{\log r})$ contains a $K_r$ minor, a partial result towards Hadwiger's famous conjecture. In this paper, we investigate analogues of these results in the directed setting. There are several ways to define a minor in a digraph. One natural way is as follows. A strong $\overrightarrow{K}_r$ minor is a digraph whose vertex set is partitioned into $r$ parts such that each part induces a strongly-connected subdigraph, and there is at least one edge in each direction between any two distinct parts. We investigate bounds on the dichromatic number and minimum out-degree of a digraph that force the existence of strong $\overrightarrow{K}_r$ minors as subdigraphs. In particular, we show that any tournament with dichromatic number at least $2r$ contains a strong $\overrightarrow{K}_r$ minor, and any tournament with minimum out-degree $Ω(r\sqrt{\log r})$ also contains a strong $\overrightarrow{K}_r$ minor. The latter result is tight up to the implied constant, and may be viewed as a strong-minor analogue to the classical result of Kostochka and Thomason. Lastly, we show that there is no function $f: \mathbb{N} \rightarrow \mathbb{N}$ such that any digraph with minimum out-degree at least $f(r)$ contains a strong $\overrightarrow{K}_r$ minor, but such a function exists when considering dichromatic number.

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arXiv2020

Bipartite independence number in graphs with bounded maximum degree

We consider a natural, yet seemingly not much studied, extremal problem in bipartite graphs. A bi-hole of size $t$ in a bipartite graph $G$ is a copy of $K_{t, t}$ in the bipartite complement of $G$. Let $f(n, Δ)$ be the largest $k$ for which every $n \times n$ bipartite graph with maximum degree $Δ$ in one of the parts has a bi-hole of size $k$. Determining $f(n, Δ)$ is thus the bipartite analogue of finding the largest independent set in graphs with a given number of vertices and bounded maximum degree. Our main result determines the asymptotic behavior of $f(n, Δ)$. More precisely, we show that for large but fixed $Δ$ and $n$ sufficiently large, $f(n, Δ) = Θ(\frac{\log Δ}Δ n)$. We further address more specific regimes of $Δ$, especially when $Δ$ is a small fixed constant. In particular, we determine $f(n, 2)$ exactly and obtain bounds for $f(n, 3)$, though determining the precise value of $f(n, 3)$ is still open.

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