arXiv subjects

Jeff Kahn

Publications and source records attributed to Jeff Kahn.

At least 37 records · Page 2Linked to original sources

arXiv2019

The number of maximal independent sets in the Hamming cube

Let $Q_n$ be the $n$-dimensional Hamming cube and $N=2^n$. We prove that the number of maximal independent sets in $Q_n$ is asymptotically \[2n2^{N/4},\] as was conjectured by Ilinca and the first author in connection with a question of Duffus, Frankl and Rödl. The value is a natural lower bound derived from a connection between maximal independent sets and induced matchings. The proof that it is also an upper bound draws on various tools, among them "stability" results for maximal independent set counts and old and new results on isoperimetric behavior in $Q_n$.

math.CO↗

arXiv2019

A Natural Extension of the BK Inequality

We extend the seminal van den Berg-Kesten Inequality on disjoint occurrence of two events to a setting with arbitrarily many events, where the quantity of interest is the maximum number that occur disjointly. This provides a handy tool for bounding upper tail probabilities for event counts in a product probability space.

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arXiv2019

The number of 4-colorings of the Hamming cube

Let $Q_d$ be the $d$-dimensional hypercube and $N=2^d$. We prove that the number of (proper) 4-colorings of $Q_d$ is asymptotically \[6e2^N,\] as was conjectured by Engbers and Galvin in 2012. The proof uses a combination of information theory (entropy) and isoperimetric ideas originating in work of Sapozhenko in the 1980's.

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arXiv2018

Stability for maximal independent sets

Answering questions of Y. Rabinovich, we prove "stability" versions of upper bounds on maximal independent set counts in graphs under various restrictions. Roughly these say that being close to the maximum implies existence of a large induced matching or triangle matching (depending on assumptions).

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arXiv2017

On regular 3-wise intersecting families

Ellis and the third author showed, verifying a conjecture of Frankl, that any $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$ admitting a transitive automorphism group has cardinality $o(2^n)$, while a construction of Frankl demonstrates that the same conclusion need not hold under the weaker constraint of being regular. Answering a question of Cameron, Frankl and Kantor from 1989, we show that the restriction of admitting a transitive automorphism group may be relaxed significantly: we prove that any $3$-wise intersecting family of subsets of $\{1,2,\dots,n\}$ that is regular and increasing has cardinality $o(2^n)$.

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arXiv2017

Disproof of a packing conjecture of Alon and Spencer

A 1992 conjecture of Alon and Spencer says, roughly, that the ordinary random graph $G_{n,1/2}$ typically admits a covering of a constant fraction of its edges by edge-disjoint, nearly maximum cliques. We show that this is not the case. The disproof is based on some (partial) understanding of a more basic question: for $k\ll \sqrt{n}$ and $A_1\dots A_t$ chosen uniformly and independently from the $k$-subsets of $\{1\dots n\}$, what can one say about \[ \mathbb{P}(|A_i\cap A_j|\leq 1 ~\forall i\neq j)? \] Our main concern is trying to understand how closely the answers to this and a related question about matchings follow heuristics gotten by pretending that certain (dependent) choices are made independently.

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arXiv2017

Proof of an entropy conjecture of Leighton and Moitra

We prove the following conjecture of Leighton and Moitra. Let $T$ be a tournament on $[n]$ and $S_n$ the set of permutations of $[n]$. For an arc $uv$ of $T$, let $A_{uv}=\{σ\in S_n \, : \, σ(u)<σ(v) \}$. $\textbf{Theorem.}$ For a fixed $\varepsilon>0$, if $\mathbb{P}$ is a probability distribution on $S_n$ such that $\mathbb{P}(A_{uv})>1/2+\varepsilon$ for every arc $uv$ of $T$, then the binary entropy of $\mathbb{P}$ is at most $(1-\vartheta_{\varepsilon})\log_2 n!$ for some (fixed) positive $\vartheta_\varepsilon$. When $T$ is transitive the theorem is due to Leighton and Moitra; for this case we give a short proof with a better $\vartheta_\varepsilon$.

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arXiv2017

Perfect fractional matchings in k-out hypergraphs

Extending the notion of (random) $k$-out graphs, we consider when the $k$-out hypergraph is likely to have a perfect fractional matching. In particular, we show that for each $r$ there is a $k=k(r)$ such that the $k$-out $r$-uniform hypergraph on $n$ vertices has a perfect fractional matching with high probability (i.e., with probability tending to $1$ as $n\to \infty$) and prove an analogous result for $r$-uniform $r$-partite hypergraphs. This is based on a new notion of hypergraph expansion and the observation that sufficiently expansive hypergraphs admit perfect fractional matchings. As a further application, we give a short proof of a stopping-time result originally due to Krivelevich.

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arXiv2016

On the Cycle Space of a Random Graph

Write $\mathcal{C}(G)$ for the cycle space of a graph $G$, $\mathcal{C}_κ(G)$ for the subspace of $\mathcal{C}(G)$ spanned by the copies of the $κ$-cycle $C_κ$ in $G$, $\mathcal{T}_κ$ for the class of graphs satisfying $\mathcal{C}_κ(G)=\mathcal{C}(G)$, and $\mathcal{Q}_κ$ for the class of graphs each of whose edges lies in a $C_κ$. We prove that for every odd $κ\geq 3$ and $G=G_{n,p}$, \[\max_p \, \Pr(G \in \mathcal{Q}_κ\setminus \mathcal{T}_κ) \rightarrow 0;\] so the $C_κ$'s of a random graph span its cycle space as soon as they cover its edges. For $κ=3$ this was shown by DeMarco, Hamm and Kahn (2013).

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arXiv2016

Chvátal's Conjecture and Correlation Inequalities

Chvátal's conjecture in extremal combinatorics asserts that for any decreasing family $\mathcal{F}$ of subsets of a finite set $S$, there is a largest intersecting subfamily of $\mathcal{F}$ consisting of all members of $\mathcal{F}$ that include a particular $x \in S$. In this paper we reformulate the conjecture in terms of influences of variables on Boolean functions and correlation inequalities, and study special cases and variants using tools from discrete Fourier analysis.

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arXiv2016

On Erdős-Ko-Rado for random hypergraphs II

Denote by $\mathcal{H}_k (n,p)$ the random $k$-graph in which each $k$-subset of $\{1... n\}$ is present with probability $p$, independent of other choices. More or less answering a question of Balogh, Bohman and Mubayi, we show: there is a fixed $\varepsilon >0$ such that if $n=2k+1$ and $p> 1-\varepsilon$, then w.h.p. (that is, with probability tending to 1 as $k\rightarrow \infty$), $\mathcal{H}_k (n,p)$ has the "Erdős-Ko-Rado property." We also mention a similar random version of Sperner's Theorem.

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arXiv2015

Tuza's Conjecture is Asymptotically Tight for Dense Graphs

An old conjecture of Zs. Tuza says that for any graph $G$, the ratio of the minimum size, $τ_3(G)$, of a set of edges meeting all triangles to the maximum size, $ν_3(G)$, of an edge-disjoint triangle packing is at most 2. Here, disproving a conjecture of R. Yuster, we show that for any fixed, positive $α$ there are arbitrarily large graphs $G$ of positive density satisfying $τ_3(G)>(1-o(1))|G|/2$ and $ν_3(G)<(1+α)|G|/4$.

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arXiv2015

On "stability" in the Erdős-Ko-Rado theorem

Denote by $K_p(n,k)$ the random subgraph of the usual Kneser graph $K(n,k)$ in which edges appear independently, each with probability $p$. Answering a question of Bollobás, Narayanan, and Raigorodskii,we show that there is a fixed $p<1$ such that a.s. (i.e., with probability tending to 1 as $k \to \infty$) the maximum independent sets of $K_p(2k+1, k)$ are precisely the sets $\{A\in V(K(2k+1,k)): x\in A\}$ ($x\in [2k+1]$). We also complete the determination of the order of magnitude of the "threshold" for the above property for general $k$ and $n\geq 2k+2$. This is new for $k\sim n/2$, while for smaller $k $ it is a recent result of Das and Tran.

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arXiv2015

Modular statistics for subgraph counts in sparse random graphs

Answering a question of Kolaitis and Kopparty, we show that, for given integer $q>1$ and pairwise nonisomorphic connected graphs $G_1...G_k$, if $p=p(n) $ is such that $\Pr(G_{n,p}\supseteq G_i)\to 1$ $\forall i$, then, with $ξ_i$ the number of copies of $G_i$ in $G_{n,p}$, $(ξ_1...ξ_k)$ is asymptotically uniformly distributed on ${\bf Z}_q^k$.

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arXiv2015

Turán's Theorem for random graphs

For a graph $G$, denote by $t_r(G)$ (resp. $b_r(G)$) the maximum size of a $K_r$-free (resp. $(r-1)$-partite) subgraph of $G$. Of course $t_r(G) \geq b_r(G)$ for any $G$, and Turán's Theorem says that equality holds for complete graphs. With $G_{n,p}$ the usual ("binomial" or "Erdős-Rényi") random graph, we show: For each fixed r there is a C such that if \[ p=p(n) > Cn^{-\tfrac{2}{r+1}}\log^{\tfrac{2}{(r+1)(r-2)}}n, \] then $\Pr(t_r(G_{n,p})=b_r(G_{n,p}))\rightarrow 1$ as $n\rightarrow\infty$. This is best possible (apart from the value of $C$) and settles a question first considered by Babai, Simonovits and Spencer about 25 years ago.

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arXiv2014

On Erdős-Ko-Rado for random hypergraphs I

A family of sets is intersecting if no two of its members are disjoint, and has the Erdős-Ko-Rado property (or is EKR) if each of its largest intersecting subfamilies has nonempty intersection. Denote by $\mathcal{H}_k(n,p)$ the random family in which each $k$-subset of $\{1\dots n\}$ is present with probability $p$, independent of other choices. A question first studied by Balogh, Bohman and Mubayi asks: \[ \mbox{for what $p=p(n,k)$ is $\mathcal{H}_k(n,p)$ likely to be EKR?} \] Here, for fixed $c<1/4$, and $k< \sqrt{cn\log n}$ we give a precise answer to this question, characterizing those sequences $p=p(n,k)$ for which $$ \Pr(\mathcal{H}_k(n,p) \textrm{ is EKR}) \rightarrow 1 \textrm{ as } n\rightarrow \infty. $$

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arXiv2014

Thresholds and expectation-thresholds of monotone properties with small minterms

Let $N$ be a finite set, let $p \in (0,1)$, and let $N_p$ denote a random binomial subset of $N$ where every element of $N$ is taken to belong to the subset independently with probability $p$ . This defines a product measure $μ_p$ on the power set of $N$, where for $\mathcal{A} \subseteq 2^N$ $μ_p(\mathcal{A}) := Pr[N_p \in \mathcal{A}]$. In this paper we study upward-closed families $\mathcal{A}$ for which all minimal sets in $\mathcal{A}$ have size at most $k$, for some positive integer $k$. We prove that for such a family $μ_p(\mathcal{A}) / p^k $ is a decreasing function, which implies a uniform bound on the coarseness of the thresholds of such families. We also prove a structure theorem which enables to identify in $\mathcal{A}$ either a substantial subfamily $\mathcal{A}_0$ for which the first moment method gives a good approximation of its measure, or a subfamily which can be well approximated by a family with all minimal sets of size strictly smaller than $k$. Finally, we relate the (fractional) expectation threshold and the probability threshold of such a family, using duality of linear programming. This is related to the threshold conjecture of Kahn and Kalai.

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arXiv2014

Influential coalitions for Boolean Functions

We improve results of Kahn, Kalai, and Linial from the late 80s on the existence of influential large coalitions for Boolean functions, and we give counterexamples to conjectures (of Benny Chor and others) also from the late 80s, by exhibiting functions for which the influences of large coalitions are unexpectedly small relative to the expectations of the functions. The large gaps between the new upper and lower bounds leave a lot of room for further study.

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