arXiv subjects
Publications and source records attributed to Liana Yepremyan.
We show that for every integer $m \ge 2$ and large $n$, every properly edge-coloured graph on $n$ vertices with at least $n (\log n)^{53}$ edges contains a rainbow subdivision of $K_m$. This is sharp up to a polylogarithmic factor. Our proof method exploits the connection between the mixing time of random walks and expansion in graphs.
For $B \subseteq \mathbb F_q^m$, the $n$-th affine extremal number of $B$ is the maximum cardinality of a set $A \subseteq \mathbb F_q^n$ with no subset which is affinely isomorphic to $B$. Furstenberg and Katznelson proved that for any $B \subseteq \mathbb F_q^m$, the $n$-th affine extremal number of $B$ is $o(q^n)$ as $n \to \infty$. By counting affine homomorphisms between subsets of $\mathbb F_q^n$, we derive new bounds and give new proofs of some previously known bounds for certain affine extremal numbers. At the same time, we establish corresponding supersaturation results. We connect these bounds to certain Ramsey-type numbers in vector spaces over finite fields. For $s,t \geq 1$, let $R_q(s,t)$ denote the minimum $n$ such that in every red-blue coloring of the one-dimensional subspaces of $\mathbb F_q^n$, there is either a red $s$-dimensional subspace or a blue $t$-dimensional subspace of $\mathbb F_q^n$. The existence of these numbers is a special case of a well-known theorem of Graham, Leeb, Rothschild. We improve the best known upper bounds on $R_2(2,t)$, $R_3(2,t)$, $R_2(t,t)$, and $R_3(t,t)$.
We show that Erdős-Rényi random graphs $G(n,p)$ with constant density $p<1$ have correspondence chromatic number $O(n/\sqrt{\log n})$; this matches a prediction from linear Hadwiger's conjecture for correspondence coloring. The proof follows from a simple sufficient condition for correspondence colorability in terms of the numbers of independent sets.
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.
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.
We show that any connected Cayley graph $Γ$ on an Abelian group of order $2n$ and degree $\tildeΩ(\log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $Γ$ is bipartite. Our proof is based on Sapozhenko's graph container method and uses the Plünnecke-Rusza-Petridis inequality from additive combinatorics.
The $s$-colour size-Ramsey number of a hypergraph $H$ is the minimum number of edges in a hypergraph $G$ whose every $s$-edge-colouring contains a monochromatic copy of $H$. We show that the $s$-colour size-Ramsey number of the $t$-power of the $r$-uniform tight path on $n$ vertices is linear in $n$, for every fixed $r, s, t$, thus answering a question of Dudek, La Fleur, Mubayi, and Rödl (2017). In fact, we prove a stronger result that allows us to deduce that powers of bounded degree hypergraph trees and powers of `long subdivisions' of bounded degree hypergraphs have size-Ramsey numbers that are linear in the number of vertices. This extends and strongly generalises recent results about the linearity of size-Ramsey numbers of powers of bounded degree trees and of long subdivisions of bounded degree graphs.
Given $r$-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 $\rm ex(G_{n,p}^{(r)}, C_{2\ell}^{(r)})$ up to polylogarithmic factors for all but a small interval of values of $p=p(n)$ whose length decreases as $\ell$ grows. Our main technical contribution is a balanced supersaturation result for linear even cycles which improves upon previous such results by Ferber-Mckinley-Samotij and Balogh-Narayanan-Skokan. The novelty is that the supersaturation result depends on the codegree of some pairs of vertices in the underlying hypergraph. This approach could be used to prove similar results for other hypergraphs $H$.
Tree-chromatic number is a chromatic version of treewidth, where the cost of a bag in a tree-decomposition is measured by its chromatic number rather than its size. Path-chromatic number is defined analogously. These parameters were introduced by Seymour (JCTB 2016). In this paper, we survey all the known results on tree- and path-chromatic number and then present some new results and conjectures. In particular, we propose a version of Hadwiger's Conjecture for tree-chromatic number. As evidence that our conjecture may be more tractable than Hadwiger's Conjecture, we give a short proof that every $K_5$-minor-free graph has tree-chromatic number at most $4$, which avoids the Four Colour Theorem. We also present some hardness results and conjectures for computing tree- and path-chromatic number.
A well-known result of Verstraëte \cite{V00} shows that for each integer $k\geq 2$ every graph $G$ with average degree at least $8k$ contains cycles of $k$ consecutive even lengths, the shortest of which is at most twice the radius of $G$. We establish two extensions of Verstraëte's result for linear cycles in linear $r$-uniform hypergraphs. We show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_1=c_1(r)$ and $c_2=c_2(r,k)$, such that every linear $r$-uniform hypergraph $G$ with average degree $d(G)\geq c_1 k$ contains linear cycles of $k$ consecutive even lengths, the shortest of which is at most $2\lceil \frac{ \log n}{\log (d(G)/k)-c_2}\rceil$. In particular, as an immediate corollary, we retrieve the current best known upper bound on the linear Turán number of $C^r_{2k}$ with improved coefficients. Furthermore, we show that for any fixed integers $r\geq 3,k\geq 2$, there exist constants $c_3=c_3(r)$ and $c_4=c_4(r)$ such that every $n$-vertex linear $r$-uniform graph with average degree $d(G)\geq c_3k$, contains linear cycles of $k$ consecutive lengths, the shortest of which has length at most $6\lceil \frac{\log n}{\log (d(G)/k)-c_4} \rceil +6$. Both the degree condition and the shortest length among the cycles guaranteed are best possible up to a constant factor.
A Latin square of order $n$ is an $n \times n$ array filled with $n$ symbols such that each symbol appears only once in every row or column and a transversal is a collection of cells which do not share the same row, column or symbol. The study of Latin squares goes back more than 200 years to the work of Euler. One of the most famous open problems in this area is a conjecture of Ryser-Brualdi-Stein from 60s which says that every Latin square of order $n\times n$ contains a transversal of order $n-1$. In this paper we prove the existence of a transversal of order $n-O(\log{n}/\log{\log{n}})$, improving the celebrated bound of $n-O(\log^2n)$ by Hatami and Shor. Our approach (different from that of Hatami-Shor) is quite general and gives several other applications as well. We obtain a new lower bound on a 40 year old conjecture of Brouwer on the maximum matching in Steiner triple systems, showing that every such system of order $n$ is guaranteed to have a matching of size $n/3-O(\log{n}/\log{\log{n}})$. This substantially improves the current best result of Alon, Kim and Spencer which has the error term of order $n^{1/2+o(1)}$. Finally, we also show that $O(n\log{n}/\log{\log{n}})$ many symbols in Latin arrays suffice to guarantee a full transversal, improving on previously known bound of $n^{2-\varepsilon}$. The proofs combine in a novel way the semirandom method together with the robust expansion properties of edge coloured pseudorandom graphs to show the existence of a rainbow matching covering all but $O(\log n/\log{\log{n}})$ vertices. All previous results, based on the semi-random method, left uncovered at least $Ω(n^α)$ (for some constant $α$) vertices.
We show that if a multigraph $G$ with maximum edge-multiplicity of at most $\frac{\sqrt{n}}{\log^2 n}$, is edge-coloured by $n$ colours such that each colour class is a disjoint union of cliques with at least $2n + o(n)$ vertices, then it has a full rainbow matching, that is, a matching where each colour appears exactly once. This asymptotically solves a question raised by Clemens, Ehrenmüller and Pokrovskiy, and is related to problems on algebras of sets studied by Grinblat in [Grinblat 2002].
An important theme of recent research in Ramsey theory has been establishing pseudorandomness properties of Ramsey graphs. An $N$-vertex graph is called $C$-Ramsey if it has no homogeneous set of size $C\log N$. A theorem of Bukh and Sudakov, solving a conjecture of Erdős, Faudree and Sós, shows that any $C$-Ramsey $N$-vertex graph contains an induced subgraph with $Ω_C(N^{1/2})$ distinct degrees. We improve this to $Ω_C(N^{2/3})$, which is tight up to the constant factor. We also show that any $N$-vertex graph with $N > (k-1)(n-1)$ and $n\geq n_0(k) = Ω(k^9)$ either contains a homogeneous set of order $n$ or an induced subgraph with $k$ distinct degrees. The lower bound on $N$ here is sharp, as shown by an appropriate Turán graph, and confirms a conjecture of Narayanan and Tomon.
The celebrated Hadwiger's conjecture states that if a graph contains no $K_{t+1}$ minor then it is $t$-colourable. If true, it would in particular imply that every $n$-vertex $K_{t+1}$-minor-free graph has an independent set of size at least $n/t$. In 1982, Duchet and Meyniel proved that this bound holds within a factor $2$. Their bound has been improved; most notably in an absolute factor by Fox, which was later improved by Balogh and Kostochka. Here we consider the same question for triangle-free graphs. By the results of Shearer and Kostochka and Thomason, it follows that any triangle-free graph with no $K_t$ minor has an independent set of size $Ω(\tfrac{\sqrt{\log{t}}}{t}n)$. We show that a much larger independent set exists; for all sufficiently large $t$ every triangle-free graph on $n$ vertices with no $K_t$-minor has an independent set of size $ \tfrac{n}{t^{1-\varepsilon}}$. This answers a question of Sergey Norin.
A graph $G$ is Ramsey for a graph $H$ if every 2-colouring of the edges of $G$ contains a monochromatic copy of $H$. We consider the following question: if $H$ has bounded treewidth, is there a `sparse' graph $G$ that is Ramsey for $H$? Two notions of sparsity are considered. Firstly, we show that if the maximum degree and treewidth of $H$ are bounded, then there is a graph $G$ with $O(|V(H)|)$ edges that is Ramsey for $H$. This was previously only known for the smaller class of graphs $H$ with bounded bandwidth. On the other hand, we prove that the treewidth of a graph $G$ that is Ramsey for $H$ cannot be bounded in terms of the treewidth of $H$ alone. In fact, the latter statement is true even if the treewidth is replaced by the degeneracy and $H$ is a tree.
A long-standing conjecture of Erdős and Simonovits asserts that for every rational number $r\in (1,2)$ there exists a bipartite graph $H$ such that $\ex(n,H)=Θ(n^r)$. So far this conjecture is known to be true only for rationals of form $1+1/k$ and $2-1/k$, for integers $k\geq 2$. In this paper we add a new form of rationals for which the conjecture is true; $2-2/(2k+1)$, for $k\geq 2$. This in its turn also gives an affirmative answer to a question of Pinchasi and Sharir on cube-like graphs. Recently, a version of Erdős and Simonovits's conjecture where one replaces a single graph by a family, was confirmed by Bukh and Conlon. They proposed a construction of bipartite graphs which should satisfy Erdős and Simonovits's conjecture. Our result can also be viewed as a first step towards verifying Bukh and Conlon's conjecture. We also prove the an upper bound on the Turán's number of $θ$-graphs in an asymmetric setting and employ this result to obtain yet another new rational exponent for Turán exponents; $r=7/5$.
Akbari and Alipour conjectured that any Latin array of order $n$ with at least $n^2/2$ symbols contains a transversal. We confirm this conjecture for large $n$, and moreover, we show that $n^{399/200}$ symbols suffice.
For any $r$-graph $H$, we consider the problem of finding a rainbow $H$-factor in an $r$-graph $G$ with large minimum $\ell$-degree and an edge-colouring that is suitably bounded. We show that the asymptotic degree threshold is the same as that for finding an $H$-factor.