arXiv subjects

Benny Sudakov

Publications and source records attributed to Benny Sudakov.

At least 217 records · Page 12Linked to original sources

arXiv2014

Judicious partitions of directed graphs

The area of judicious partitioning considers the general family of partitioning problems in which one seeks to optimize several parameters simultaneously, and these problems have been widely studied in various combinatorial contexts. In this paper, we study essentially the most fundamental judicious partitioning problem for directed graphs, which naturally extends the classical Max Cut problem to this setting: we seek bipartitions in which many edges cross in each direction. It is easy to see that a minimum outdegree condition is required in order for the problem to be nontrivial, and we prove that every directed graph with M edges and minimum outdegree at least two admits a bipartition in which at least (1/6 + o(1))M edges cross in each direction. We also prove that if the minimum outdegree is at least three, then the constant can be increased to 1/5. If the minimum outdegree tends to infinity with N, then the constant increases to 1/4. All of these constants are best-possible, and provide asymptotic answers to a question of Alex Scott.

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arXiv2014

On the grid Ramsey problem and related questions

The Hales--Jewett theorem is one of the pillars of Ramsey theory, from which many other results follow. A celebrated theorem of Shelah says that Hales--Jewett numbers are primitive recursive. A key tool used in his proof, now known as the cube lemma, has become famous in its own right. In its simplest form, this lemma says that if we color the edges of the Cartesian product $K_n \times K_n$ in $r$ colors then, for $n$ sufficiently large, there is a rectangle with both pairs of opposite edges receiving the same color. Shelah's proof shows that $n = r^{\binom{r+1}{2}} + 1$ suffices. More than twenty years ago, Graham, Rothschild and Spencer asked whether this bound can be improved to a polynomial in $r$. We show that this is not possible by providing a superpolynomial lower bound in $r$. We also discuss a number of related problems.

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arXiv2014

The threshold probability for long cycles

For a given graph $G$ of minimum degree at least $k$, let $G_p$ denote the random spanning subgraph of $G$ obtained by retaining each edge independently with probability $p=p(k)$. We prove that if $p \ge \frac{\log k + \log \log k + ω_k(1)}{k}$, where $ω_k(1)$ is any function tending to infinity with $k$, then $G_p$ asymptotically almost surely contains a cycle of length at least $k+1$. When we take $G$ to be the complete graph on $k+1$ vertices, our theorem coincides with the classic result on the threshold probability for the existence of a Hamilton cycle in the binomial random graph.

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arXiv2014

The minimum number of nonnegative edges in hypergraphs

An r-unform n-vertex hypergraph H is said to have the Manickam-Miklós-Singhi (MMS) property if for every assignment of weights to its vertices with nonnegative sum, the number of edges whose total weight is nonnegative is at least the minimum degree of H. In this paper we show that for n>10r^3, every r-uniform n-vertex hypergraph with equal codegrees has the MMS property, and the bound on n is essentially tight up to a constant factor. This result has two immediate corollaries. First it shows that every set of n>10k^3 real numbers with nonnegative sum has at least $\binom{n-1}{k-1}$ nonnegative k-sums, verifying the Manickam-Miklós-Singhi conjecture for this range. More importantly, it implies the vector space Manickam-Miklós-Singhi conjecture which states that for n >= 4k and any weighting on the 1-dimensional subspaces of F_q^n with nonnegative sum, the number of nonnegative k-dimensional subspaces is at least ${n-1 \brack k-1}_q$. We also discuss two additional generalizations, which can be regarded as analogues of the Erdős-Ko-Rado theorem on k-intersecting families.

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arXiv2014

On the number of monotone sequences

One of the most classical results in Ramsey theory is the theorem of Erdős and Szekeres from 1935, which says that every sequence of more than $k^2$ numbers contains a monotone subsequence of length $k+1$. We address the following natural question motivated by this result: Given integers $k$ and $n$ with $n \geq k^2+1$, how many monotone subsequences of length $k+1$ must every sequence of $n$ numbers contain? We answer this question precisely for all sufficiently large $k$ and $n \leq k^2 + c k^{3/2} / \log k$, where $c$ is some absolute positive constant.

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arXiv2014

Cycle packing

In the 1960s, Erdős and Gallai conjectured that the edge set of every graph on n vertices can be partitioned into O(n) cycles and edges. They observed that one can easily get an O(n log n) upper bound by repeatedly removing the edges of the longest cycle. We make the first progress on this problem, showing that O(n log log n) cycles and edges suffice. We also prove the Erdős-Gallai conjecture for random graphs and for graphs with linear minimum degree.

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arXiv2014

Random directed graphs are robustly Hamiltonian

A classical theorem of Ghouila-Houri from 1960 asserts that every directed graph on $n$ vertices with minimum out-degree and in-degree at least $n/2$ contains a directed Hamilton cycle. In this paper we extend this theorem to a random directed graph ${\mathcal D}(n,p)$, that is, a directed graph in which every ordered pair $(u,v)$ becomes an arc with probability $p$ independently of all other pairs. Motivated by the study of resilience of properties of random graphs, we prove that if $p \gg \log n/\sqrt{n}$, then a.a.s. every subdigraph of ${\mathcal D}(n,p)$ with minimum out-degree and in-degree at least $(1/2 + o(1)) n p$ contains a directed Hamilton cycle. The constant $1/2$ is asymptotically best possible. Our result also strengthens classical results about the existence of directed Hamilton cycles in random directed graphs.

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arXiv2014

Cycles in triangle-free graphs of large chromatic number

More than twenty years ago Erdős conjectured~\cite{E1} that a triangle-free graph $G$ of chromatic number $k \geq k_0(\varepsilon)$ contains cycles of at least $k^{2 - \varepsilon}$ different lengths as $k \rightarrow \infty$. In this paper, we prove the stronger fact that every triangle-free graph $G$ of chromatic number $k \geq k_0(\varepsilon)$ contains cycles of $(\frac{1}{64} - \varepsilon)k^2 \log k$ consecutive lengths, and a cycle of length at least $(\tfrac{1}{4} - \varepsilon)k^2 \log k$. As there exist triangle-free graphs of chromatic number $k$ with at most roughly $4k^2 \log k$ vertices for large $k$, theses results are tight up to a constant factor. We also give new lower bounds on the circumference and the number of different cycle lengths for $k$-chromatic graphs in other monotone classes, in particular, for $K_r$-free graphs and graphs without odd cycles $C_{2s+1}$.

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arXiv2014

The Erdős-Gyárfás problem on generalized Ramsey numbers

Fix positive integers $p$ and $q$ with $2 \leq q \leq {p \choose 2}$. An edge-coloring of the complete graph $K_n$ is said to be a $(p, q)$-coloring if every $K_p$ receives at least $q$ different colors. The function $f(n, p, q)$ is the minimum number of colors that are needed for $K_n$ to have a $(p,q)$-coloring. This function was introduced by Erdős and Shelah about 40 years ago, but Erdős and Gyárfás were the first to study the function in a systematic way. They proved that $f(n, p, p)$ is polynomial in $n$ and asked to determine the maximum $q$, depending on $p$, for which $f(n,p,q)$ is subpolynomial in $n$. We prove that the answer is $p-1$.

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arXiv2014

Cores of random graphs are born Hamiltonian

Let $(G_t)_{t \geq 0}$ be the random graph process ($G_0$ is edgeless and $G_t$ is obtained by adding a uniformly distributed new edge to $G_{t-1}$), and let $τ_k$ denote the minimum time $t$ such that the $k$-core of $G_t$ (its unique maximal subgraph with minimum degree at least $k$) is nonempty. For any fixed $k\geq 3$ the $k$-core is known to emerge via a discontinuous phase transition, where at time $t=τ_k$ its size jumps from 0 to linear in the number of vertices with high probability. It is believed that for any $k\geq 3$ the core is Hamiltonian upon creation w.h.p., and Bollobás, Cooper, Fenner and Frieze further conjectured that it in fact admits $\lfloor(k-1)/2\rfloor$ edge-disjoint Hamilton cycles. However, even the asymptotic threshold for Hamiltonicity of the $k$-core in $G(n,p)$ was unknown for any $k$. We show here that for any fixed $k\ge 15$ the $k$-core of $G_t$ is w.h.p. Hamiltonian for all $t \geq τ_k$, i.e., immediately as the $k$-core appears and indefinitely afterwards. Moreover, we prove that for large enough fixed $k$ the $k$-core contains $\lfloor (k-3)/2\rfloor$ edge-disjoint Hamilton cycles w.h.p. for all $t\geq τ_k$.

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arXiv2014

Large subgraphs without complete bipartite graphs

In this note, we answer the following question of Foucaud, Krivelevich and Perarnau. What is the size of the largest $K_{r,s}$-free subgraph one can guarantee in every graph $G$ with $m$ edges? We also discuss the analogous problem for hypergraphs.

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arXiv2014

Maximizing the number of independent sets of a fixed size

Let $i_t(G)$ be the number of independent sets of size $t$ in a graph $G$. Engbers and Galvin asked how large $i_t(G)$ could be in graphs with minimum degree at least $δ$. They further conjectured that when $n\geq 2δ$ and $t\geq 3$, $i_t(G)$ is maximized by the complete bipartite graph $K_{δ, n-δ}$. This conjecture has drawn the attention of many researchers recently. In this short note, we prove this conjecture.

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arXiv2013

Ramsey numbers of cubes versus cliques

The cube graph Q_n is the skeleton of the n-dimensional cube. It is an n-regular graph on 2^n vertices. The Ramsey number r(Q_n, K_s) is the minimum N such that every graph of order N contains the cube graph Q_n or an independent set of order s. Burr and Erdos in 1983 asked whether the simple lower bound r(Q_n, K_s) >= (s-1)(2^n - 1)+1 is tight for s fixed and n sufficiently large. We make progress on this problem, obtaining the first upper bound which is within a constant factor of the lower bound.

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arXiv2013

On the 3-local profiles of graphs

For a graph G, let p_i(G), i=0,...,3 be the probability that three distinct random vertices span exactly i edges. We call (p_0(G),...,p_3(G)) the 3-local profile of G. We investigate the set ${\cal S}_3 \subset \mathbb R^4$ of all vectors (p_0,...,p_3) that are arbitrarily close to the 3-local profiles of arbitrarily large graphs. We give a full description of the projection of ${\cal S}_3$ to the (p_0, p_3) plane. The upper envelope of this planar domain is obtained from cliques on a fraction of the vertex set and complements of such graphs. The lower envelope is Goodman's inequality p_0+p_3\geq 1/4. We also give a full description of the triangle-free case, i.e., the intersection of ${\cal S}_3$ with the hyperplane p_3=0. This planar domain is characterized by an SDP constraint that is derived from Razborov's flag algebra theory.

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arXiv2013

On the densities of cliques and independent sets in graphs

Let r, s >= 2 be integers. Suppose that the number of blue r-cliques in a red/blue coloring of the edges of the complete graph K_n is known and fixed. What is the largest possible number of red s-cliques under this assumption? The well known Kruskal-Katona theorem answers this question for r=2 or s=2. Using the shifting technique from extremal set theory together with some analytical arguments, we resolve this problem in general and prove that in the extremal coloring either the blue edges or the red edges form a clique.

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arXiv2013

Two extensions of Ramsey's theorem

Ramsey's theorem, in the version of Erdős and Szekeres, states that every 2-coloring of the edges of the complete graph on {1, 2,...,n} contains a monochromatic clique of order 1/2\log n. In this paper, we consider two well-studied extensions of Ramsey's theorem. Improving a result of Rödl, we show that there is a constant $c>0$ such that every 2-coloring of the edges of the complete graph on \{2, 3,...,n\} contains a monochromatic clique S for which the sum of 1/\log i over all vertices i \in S is at least c\log\log\log n. This is tight up to the constant factor c and answers a question of Erdős from 1981. Motivated by a problem in model theory, Väänänen asked whether for every k there is an n such that the following holds. For every permutation πof 1,...,k-1, every 2-coloring of the edges of the complete graph on {1, 2, ..., n} contains a monochromatic clique a_1<... a_{π(2)+1}-a_{π(2)}>...>a_{π(k-1)+1}-a_{π(k-1)}. That is, not only do we want a monochromatic clique, but the differences between consecutive vertices must satisfy a prescribed order. Alon and, independently, Erdős, Hajnal and Pach answered this question affirmatively. Alon further conjectured that the true growth rate should be exponential in k. We make progress towards this conjecture, obtaining an upper bound on n which is exponential in a power of k. This improves a result of Shelah, who showed that n is at most double-exponential in k.

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arXiv2013

Short proofs of some extremal results

We prove several results from different areas of extremal combinatorics, giving complete or partial solutions to a number of open problems. These results, coming from areas such as extremal graph theory, Ramsey theory and additive combinatorics, have been collected together because in each case the relevant proofs are quite short.

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arXiv2013

The minimum number of disjoint pairs in set systems and related problems

Let F be a set system on [n] with all sets having k elements and every pair of sets intersecting. The celebrated theorem of Erdos-Ko-Rado from 1961 says that any such system has size at most ${n-1 \choose k-1}$. A natural question, which was asked by Ahlswede in 1980, is how many disjoint pairs must appear in a set system of larger size. Except for the case k=2, solved by Ahlswede and Katona, this problem has remained open for the last three decades. In this paper, we determine the minimum number of disjoint pairs in small k-uniform families, thus confirming a conjecture of Bollobas and Leader in these cases. Moreover, we obtain similar results for two well-known extensions of the Erdos-Ko-Rado theorem, determining the minimum number of matchings of size q and the minimum number of t-disjoint pairs that appear in set systems larger than the corresponding extremal bounds. In the latter case, this provides a partial solution to a problem of Kleitman and West.

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