arXiv subjects

Raphael Yuster

Publications and source records attributed to Raphael Yuster.

At least 19 recordsLinked to original sources

arXiv2026

A Tight Erd\H{o}s-Stone Bound for All Graph Densities

The Erd\H{o}s--Stone Theorem asserts that if a graph has edge density $1-1/r+\delta$ then it contains a complete $(r+1)$-partite graph with $b$ vertices in each part, where $b=b_n(r,\delta) \gg 1$. The celebrated Chv\'atal--Szemer\'edi theorem determined the exact order of $b_n(r,\delta)$ for every $\delta < 1/r^3$. Their bound, however, is not tight when $\delta=1/r-\epsilon$, that is, when the graph has edge density $1-\epsilon$ for small $\epsilon$. Our main result in this paper determines the correct order in this remaining regime, thereby enabling us to give a tight bound for the Erd\H{o}s--Stone problem for all edge densities. More precisely, we prove that for every integer $r\geq 2$ and $0< \delta < 1/r$ we have $$ b_n(r,\delta)=\Theta\left(\frac{\log n}{(1/r-\delta)r\log(1/\delta)}\right)\;. $$ The lower bound is obtained using a K\"{o}vari-S\'os-Tur\'an-type argument combined with a variant of Nikiforov's method of constructing large blow-ups, while the upper bound is proved using a correlated random graph construction, related to tensor powers.

math.CO↗

arXiv2026

On Ramsey Properties of k-Majority Tournaments

A central objective in Ramsey theory is determining whether restricted families of discrete structures necessarily contain substantially larger homogeneous substructures, compared to the unrestricted structures. In the setting of tournaments, it is well known that every tournament contains a transitive subgraph of size $\log n$, and that this is best possible up to a constant factor. A restricted family of tournaments that has been extensively studied is the family of $k$-majority tournaments. They are obtained by taking $2k-1$ linear orders of a set $X$, and defining a tournament on $X$ which has an edge from $u$ to $v$ if $u$ precedes $v$ in at least $k$ of these orders. Milans, Schreiber, and West proved that such tournaments indeed have significantly larger transitive tournaments. More precisely, they proved that every $k$-majority tournament contains a transitive tournament of size $n^{2^{-\Theta(k)}}$. Our main goal in this paper is to give an exponential improvement in the dependence of the exponent on $k$ by showing that every $k$-majority tournament contains a transitive set of size $n^{\Omega(1/k)}$. Finally, we highlight several open problems and conjectural directions related to random $k$-majority tournaments.

math.CO↗

arXiv2026

Simultaneous separation in bounded degree trees

It follows from a classical result of Jordan that every tree with maximum degree at most $r$ containing a vertex set labeled by $[n]$, has a single-edge cut which separates two subsets $A,B \subset [n]$ for which $\min\{|A|,|B|\} \ge (n-1)/r$. Motivated by the tree dissimilarity problem in phylogenetics, we consider the case of separating vertex sets of {\em several} trees: Given $k$ trees with maximum degree at most $r$, containing a common vertex set labeled by $[n]$, we ask for a single-edge cut in each tree which maximizes $min\{|A|,|B|\}$ where $A,B \subset [n]$ are separated by the corresponding cut at each tree. Denoting this maximum by $f(r,k,n)$ and considering the limit $f(r,k) = \lim_{n \rightarrow \infty} f(r,k,n)/n$ (which is shown to always exist) we determine that $f(r,2)=\frac{1}{2r}$ and determine that $f(3,3)=\frac{2}{27}$, which is already quite intricate. The case $r=3$ is especially interesting in phylogenetics and our result implies that any two (three) binary phylogenetic trees over $n$ taxa have a split at each tree which separates two taxa sets of order at least $n/6$ (resp. $2n/27$), and these bounds are asymptotically tight.

math.CO↗

arXiv2026

On the maximum density of a matrix and a transcendental Tur\'an-type density

We prove that the inducibility of $P_4$ in ordered monotone balanced bipartite graphs is $2/e^2$, establishing the smallest known graph with transcendental Tur\'an-type density. Moreover, the limit object is a binary graphon, so it generates a deterministic model. This is a special case of a more general framework addressed here -- the asymptotic maximum density of a constant matrix over an arbitrary symbol set, in a large, possibly monotone, matrix. We solve all $2 \times 2$ monotone cases (one of which corresponds to the aforementioned $P_4$) and all but one of the $2 \times 2$ unrestricted cases. While $(h!/h^h)^2$ is a lower bound for the asymptotic maximum density of an $h \times h$ matrix, we explicitly construct, for all $h \ge 1$, an $h \times h$ minimizer, i.e., a matrix for which this bound is attained. We also sketch how known results on the inducibility of graphs can be modified to show that, as $h$ grows, almost all $h \times h$ $0/1$ matrices are minimizers.

math.CO↗

arXiv2025

Acyclic subgraphs of digraphs with high chromatic number

For a digraph $G$, let $f(G)$ be the maximum chromatic number of an acyclic subgraph of $G$. For an $n$-vertex digraph $G$ it is proved that $f(G) \ge n^{5/9-o(1)}s^{-14/9}$ where $s$ is the bipartite independence number of $G$, i.e., the largest $s$ for which there are two disjoint $s$-sets of vertices with no edge between them. This generalizes a result of Fox, Kwan and Sudakov, who proved this for the case $s=0$ (i.e., tournaments and semicomplete digraphs). Consequently, if $s=n^{o(1)}$, then $f(G) \ge n^{5/9-o(1)}$ which polynomially improves the folklore bound $f(G) \ge n^{1/2-o(1)}$. As a corollary, with high probability, all orientations of the random $n$-vertex graph with edge probability $p=n^{-o(1)}$ (in particular, constant $p$, hence almost all $n$-vertex graphs) satisfy $f(G) \ge n^{5/9-o(1)}$. Our proof uses a theorem of Gallai and Milgram that together with several additional ideas, essentially reduces to the proof of Fox, Kwan and Sudakov.

math.CO↗

arXiv2025

Inducibility in $H$-free graphs and inducibility of Tur\'an graphs

For graphs $F$ and $H$, let $i(F)$ denote the inducibility of $F$ and let $i_H(F)$ denote the inducibility of $F$ over $H$-free graphs. We prove that for almost all graphs $F$ on a given number of vertices, $i_{K_k}(F)$ attains infinitely many values as $k$ varies. For complete partite graphs $F$ (and, more generally, for symmetrizable families of graphs $F$), we prove that $i_H(F)=i_{K_k}(F)$ where $k=\chi(H)$, and is attained by a complete $\ell$-partite graphon $W_{F,k}$, where $\ell < k$. We determine the part sizes of $W_{F,k}$ for all $k$, whence determine $i(F)$, whenever $F$ is the Tur\'an graph on $s$ vertices and $r$ parts, for all $s \le 3r+1$, which was recently proved by Liu, Mubayi, and Reiher for $s=r+1$. As a corollary, this determines the inducibility of all Tur\'an graphs on at most $14$ vertices. Furthermore, since inducibility is invariant under complement, this determines the inducibility of all matchings and, more generally, all graphs with maximum degree $1$, of any size. Similarly, this determines the inducibility of all triangle factors, of any size. For complete partite graphs $F$ with at most one singleton part, we prove that $i_{K_k}(F)$ only attains finitely many values as $k$ varies; in particular, there exists $t=t(F)$ such that $i(F)$ is attained by some complete $t$-partite graphon. This is best possible as it was shown by Liu, Pikhurko, Sharifzadeh, and Staden that this is not necessarily true if there are two singleton parts. Finally, for every $r$, we give a nontrivial sufficient condition for a complete $r$-partite graph $F$ to have the property that $i(F)$ is attained by a complete partite graphon all whose part sizes are distinct.

math.CO↗

arXiv2024

On the minimum density of monotone subwords

We consider the asymptotic minimum density $f(s,k)$ of monotone $k$-subwords of words over a totally ordered alphabet of size $s$. The unrestricted alphabet case, $f(\infty,k)$, is well-studied, known for $f(\infty,3)$ and $f(\infty,4)$, and, in particular, conjectured to be rational for all $k$. Here we determine $f(2,k)$ for all $k$ and determine $f(3,3)$, which is already irrational. We describe an explicit construction for all $s$ which is conjectured to yield $f(s,3)$. Using our construction and flag algebra, we determine $f(4,3),f(5,3),f(6,3)$ up to $10^{-3}$ yet argue that flag algebra, regardless of computational power, cannot determine $f(5,3)$ precisely. Finally, we prove that for every fixed $k \ge 3$, the gap between $f(s,k)$ and $f(\infty,k)$ is $\Theta(\frac{1}{s})$.

math.CO↗

arXiv2024

Path-monochromatic bounded depth rooted trees in (random) tournaments

An edge-colored rooted directed tree (aka arborescence) is path-monochromatic if every path in it is monochromatic. Let $k,\ell$ be positive integers. For a tournament $T$, let $f_T(k)$ be the largest integer such that every $k$-edge coloring of $T$ has a path-monochromatic subtree with at least $f_T(k)$ vertices and let $f_T(k,\ell)$ be the restriction to subtrees of depth at most $\ell$. It was proved by Landau that $f_T(1,2)=n$ and proved by Sands et al. that $f_T(2)=n$ where $|V(T)|=n$. Here we consider $f_T(k)$ and $f_T(k,\ell)$ in more generality, determine their extremal values in most cases, and in fact in all cases assuming the Caccetta-H\"aggkvist Conjecture. We also study the typical value of $f_T(k)$ and $f_T(k,\ell)$, i.e., when $T$ is a random tournament.

math.CO↗

arXiv2024

Highly connected graphs have highly connected spanning bipartite subgraphs

For integers $k,n$ with $1 \le k \le n/2$, let $f(k,n)$ be the smallest integer $t$ such that every $t$-connected $n$-vertex graph has a spanning bipartite $k$-connected subgraph. A conjecture of Thomassen asserts that $f(k,n)$ is upper bounded by some function of $k$. The best upper bound for $f(k,n)$ is by Delcourt and Ferber who proved that $f(k,n) \le 10^{10}k^3 \log n$. Here it is proved that $f(k,n) \le 22k^2 \log n$. For larger $k$, stronger bounds hold. In the linear regime, it is proved that for any $0 < c < \frac{1}{2}$ and all sufficiently large $n$, if $k=\lfloor cn \rfloor$, then $f(k, n) \le 30\sqrt{c} n \le 30\sqrt{n(k+1)}$. In the polynomial regime, it is proved that for any $\frac{1}{3} \le \alpha < 1$ and all sufficiently large $n$, if $k = \lfloor n^\alpha \rfloor$, then $f(k ,n) \le 9n^{(1+\alpha)/2} \le 9\sqrt{n(k+1)}$.

math.CO↗

arXiv2023

Flip colouring of graphs

It is proved that for integers $b, r$ such that $3 \leq b < r \leq \binom{b+1}{2} - 1$, there exists a red/blue edge-colored graph such that the red degree of every vertex is $r$, the blue degree of every vertex is $b$, yet in the closed neighborhood of every vertex there are more blue edges than red edges. The upper bound $r \le \binom{b+1}{2}-1$ is best possible for any $b \ge 3$. We further extend this theorem to more than two colours, and to larger neighbourhoods. A useful result required in some of our proofs, of independent interest, is that for integers $r,t$ such that $0 \leq t \le \frac{r^2}{2} - 5r^{3/2}$, there exists an $r$-regular graph in which each open neighborhood induces precisely $t$ edges. Several explicit constructions are introduced and relationships with constant linked graphs, $(r,b)$-regular graphs and vertex transitive graphs are revealed.

math.CO↗

arXiv2023

Packing and covering a given directed graph in a directed graph

For every fixed $k \ge 4$, it is proved that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint directed $k$-cycles, then there exists a set of at most $\frac{2}{3}kt+ o(n^2)$ arcs that meets all directed $k$-cycles and that the set of $k$-cycles admits a fractional cover of value at most $\frac{2}{3}kt$. It is also proved that the ratio $\frac{2}{3}k$ cannot be improved to a constant smaller than $\frac{k}{2}$. For $k=5$ the constant $2k/3$ is improved to $25/8$ and for $k=3$ it was recently shown by Cooper et al. that the constant can be taken to be $9/5$. The result implies a deterministic polynomial time $\frac{2}{3}k$-approximation algorithm for the directed $k$-cycle cover problem, improving upon a previous $(k{-}1)$-approximation algorithm of Kortsarz et al. More generally, for every directed graph $H$ we introduce a graph parameter $f(H)$ for which it is proved that if an $n$-vertex directed graph has at most $t$ pairwise arc-disjoint $H$-copies, then there exists a set of at most $f(H)t+ o(n^2)$ arcs that meets all $H$-copies and that the set of $H$-copies admits a fractional cover of value at most $f(H)t$. It is shown that for almost all $H$ it holds that $f(H) \approx |E(H)|/2$ and that for every $k$-vertex tournament $H$ it holds that $f(H) \le \lfloor k^2/4 \rfloor$.

math.CO↗

arXiv2023

On tournament inversion

An {\it inversion} of a tournament $T$ is obtained by reversing the direction of all edges with both endpoints in some set of vertices. Let ${\rm inv}_k(T)$ be the minimum length of a sequence of inversions using sets of size at most $k$ that result in the transitive tournament. Let ${\rm inv}_k(n)$ be the maximum of ${\rm inv}_k(T)$ taken over $n$-vertex tournaments. It is well-known that ${\rm inv}_2(n)=(1+o(1))n^2/4$ and it was recently proved by Alon et al. that ${\rm inv}(n):={\rm inv}_{n}(n)=n(1+o(1))$. In these two extreme cases ($k=2$ and $k=n$), random tournaments are asymptotically extremal objects. It is proved that the random tournament {\em does not} asymptotically attain ${\rm inv}_k(n)$ when $k \ge k_0$ and conjectured that ${\rm inv}_3(n)$ is (only) attained by (quasi) random tournaments. It is further proved that $(1+o(1)){\rm inv}_3(n)/n^2 \in [\frac{1}{12}, 0.0992)$ and $(1+o(1)){\rm inv}_k(n)/n^2 \in [\frac{1}{2k(k-1)}+\delta_k, \frac{1}{2 \lfloor k^2/2 \rfloor}-\epsilon_k]$ where $\epsilon_k > 0$ for all $k \ge 3$ and $\delta_k > 0$ for all $k \ge k_0$.

math.CO↗

arXiv2023

Finding and counting small tournaments in large tournaments

We present new algorithms for counting and detecting small tournaments in a given tournament. In particular, it is proved that every tournament on four vertices (there are four) can be detected in $O(n^2)$ time and counted in $O(n^\omega)$ time where $\omega < 2.373$ is the matrix multiplication exponent. It is also proved that any tournament on five vertices (there are $12$) can be counted in $O(n^{\omega+1})$ time. As for lower-bounds, we prove that for almost all $k$-vertex tournaments, the complexity of the detection problem is not easier than the complexity of the corresponding well-studied counting problem for {\em undirected cliques} of order $k-O(\log k)$.

cs.DS↗

arXiv2023

Almost $k$-union closed set systems

In a recent breakthrough, Gilmer proved the union closed conjecture up to a constant factor. Using Gilmer's method and additional ideas, Chase and Lovett proved an optimal result for almost union-closed set systems. Here that result is extended to higher order unions.

math.CO↗

arXiv2022

The number of bounded-degree spanning trees

For a graph $G$, let $c_k(G)$ be the number of spanning trees of $G$ with maximum degree at most $k$. For $k \ge 3$, it is proved that every connected $n$-vertex $r$-regular graph $G$ with $r \ge \frac{n}{k+1}$ satisfies $$ c_k(G)^{1/n} \ge (1-o_n(1)) r \cdot z_k $$ where $z_k > 0$ approaches $1$ extremely fast (e.g. $z_{10}=0.999971$). The minimum degree requirement is essentially tight as for every $k \ge 2$ there are connected $n$-vertex $r$-regular graphs $G$ with $r=\lfloor n/(k+1) \rfloor -2$ for which $c_k(G)=0$. Regularity may be relaxed, replacing $r$ with the geometric mean of the degree sequence and replacing $z_k$ with $z_k^* > 0$ that also approaches $1$, as long as the maximum degree is at most $n(1-(3+o_k(1))\sqrt{\ln k/k})$. The same holds with no restriction on the maximum degree as long as the minimum degree is at least $\frac{n}{k}(1+o_k(1))$.

math.CO↗

arXiv2022

The covering threshold of a directed acyclic graph by directed acyclic subgraphs

Let $H$ be a directed acyclic graph other than a rooted star. It is known that there are constants $c(H)$ and $C(H)$ such that the following holds for the complete directed graph $D_n$. There are at most $C\log n$ directed acyclic subgraphs of $D_n$ that cover every $H$-copy of $D_n$, while fewer than $c\log n$ directed acyclic subgraphs of $D_n$ do not cover all $H$-copies. Here this dichotomy is considerably strengthened. Let ${\vec G}(n,p)$ denote the random directed graph. The {\em fractional arboricity} of $H$ is $a(H) = max \{\frac{|E(H')|}{|V(H')|-1}\}$, where the maximum is over all non-singleton subgraphs of $H$. If $a(H) = \frac{|E(H)|}{|V(H)|-1}$ then $H$ is {\em totally balanced}. Complete graphs, complete multipartite graphs, cycles, trees, and, in fact, almost all graphs, are totally balanced. It is proved: 1) Let $H$ be a dag with $h$ vertices and $m$ edges other than a rooted star. For every $a^* > a(H)$ there exists $c^* = c^*(a^*,H) > 0$ such that almost surely $G \sim {\vec G}(n,n^{-1/a^*})$ has the property that every set $X$ of at most $c^*\log n$ directed acyclic subgraphs of $G$ does not cover all $H$-copies of $G$. Moreover, there exists $s(H) = m/2 + O(m^{4/5}h^{1/5})$ such that the following stronger assertion holds for any such $X$: There is an $H$-copy in $G$ that has no more than $s(H)$ of its edges covered by each element of $X$. 2) If $H$ is totally balanced then for every $0 < a^* < a(H)$, almost surely $G \sim {\vec G}(n,n^{-1/a^*})$ has a single directed acyclic subgraph that covers all its $H$-copies. As for the first result, note that if $h=o(m)$ then $s(H)=(1+o_m(1))m/2$ is about half of the edges of $H$. In fact, for infinitely many $H$ it holds that $s(H)=m/2$, optimally. As for the second result, the requirement that $H$ is totally balanced cannot, generally, be relaxed.

math.CO↗

arXiv2022

Hamiltonian cycles above expectation in r-graphs and quasi-random r-graphs

Let $H_r(n,p)$ denote the maximum number of Hamiltonian cycles in an $n$-vertex $r$-graph with density $p \in (0,1)$. The expected number of Hamiltonian cycles in the random $r$-graph model $G_r(n,p)$ is $E(n,p)=p^n(n-1)!/2$ and in the random graph model $G_r(n,m)$ with $m=p\binom{n}{r}$ it is, in fact, slightly smaller than $E(n,p)$. For graphs, $H_2(n,p)$ is proved to be only larger than $E(n,p)$ by a polynomial factor and it is an open problem whether a quasi-random graph with density $p$ can be larger than $E(n,p)$ by a polynomial factor. For hypergraphs (i.e. $r \ge 3$) the situation is drastically different. For all $r \ge 3$ it is proved that $H_r(n,p)$ is larger than $E(n,p)$ by an {\em exponential} factor and, moreover, there are quasi-random $r$-graphs with density $p$ whose number of Hamiltonian cycles is larger than $E(n,p)$ by an exponential factor.

math.CO↗

arXiv2021

On the quartet distance given partial information

Let $T$ be an arbitrary phylogenetic tree with $n$ leaves. It is well-known that the average quartet distance between two assignments of taxa to the leaves of $T$ is $\frac 23 \binom{n}{4}$. However, a longstanding conjecture of Bandelt and Dress asserts that $(\frac 23 +o(1))\binom{n}{4}$ is also the {\em maximum} quartet distance between two assignments. While Alon, Naves, and Sudakov have shown this indeed holds for caterpillar trees, the general case of the conjecture is still unresolved. A natural extension is when partial information is given: the two assignments are known to coincide on a given subset of taxa. The partial information setting is biologically relevant as the location of some taxa (species) in the phylogenetic tree may be known, and for other taxa it might not be known. What can we then say about the average and maximum quartet distance in this more general setting? Surprisingly, even determining the {\em average} quartet distance becomes a nontrivial task in the partial information setting and determining the maximum quartet distance is even more challenging, as these turn out to be dependent of the structure of $T$. In this paper we prove nontrivial asymptotic bounds that are sometimes tight for the average quartet distance in the partial information setting. We also show that the Bandelt and Dress conjecture does not generally hold under the partial information setting. Specifically, we prove that there are cases where the average and maximum quartet distance substantially differ.

q-bio.PE↗