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Noam Lifshitz

Publications and source records attributed to Noam Lifshitz.

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arXiv2018

A Note on Large H-Intersecting Families

A family $F$ of graphs on a fixed set of $n$ vertices is called triangle-intersecting if for any $G_1,G_2 \in F$, the intersection $G_1 \cap G_2$ contains a triangle. More generally, for a fixed graph $H$, a family $F$ is $H$-intersecting if the intersection of any two graphs in $F$ contains a sub-graph isomorphic to $H$. In [D. Ellis, Y. Filmus, and E. Friedgut, Triangle-intersecting families of graphs, J. Eur. Math. Soc. 14 (2012), pp. 841--885], Ellis, Filmus and Friedgut proved a 36-year old conjecture of Simonovits and Sós stating that the maximal size of a triangle-intersecting family is $(1/8)2^{n(n-1)/2}$. Furthermore, they proved a $p$-biased generalization, stating that for any $p \leq 1/2$, we have $μ_{p}(F)\le p^{3}$, where $μ_{p}(F)$ is the probability that the random graph $G(n,p)$ belongs to $F$. In the same paper, Ellis et al. conjectured that the assertion of their biased theorem holds also for $1/2 < p \le 3/4$, and more generally, that for any non-$t$-colorable graph $H$ and any $H$-intersecting family $F$, we have $μ_{p}(F)\le p^{t(t+1)/2}$ for all $p \leq (2t-1)/(2t)$. In this note we construct, for any fixed $H$ and any $p>1/2$, an $H$-intersecting family $F$ of graphs such that $μ_{p}(F)\ge 1-e^{-n^{2}/C}$, where $C$ depends only on $H$ and $p$, thus disproving both conjectures.

math.CO↗
arXiv2018

Stability versions of Erdős-Ko-Rado type theorems, via isoperimetry

Erdős-Ko-Rado (EKR) type theorems yield upper bounds on the sizes of families of sets, subject to various intersection requirements on the sets in the family. Stability versions of such theorems assert that if the size of a family is close to the maximum possible size, then the family itself must be close (in some appropriate sense) to a maximum-sized family. In this paper, we present an approach to obtaining stability versions of EKR-type theorems, via isoperimetric inequalities for subsets of the hypercube. Our approach is rather general, and allows the leveraging of a wide variety of exact EKR-type results into strong stability versions of these results, without going into the proofs of the original results. We use this approach to obtain tight stability versions of the EKR theorem itself and of the Ahlswede-Khachatrian theorem on $t$-intersecting families of $k$-element subsets of $\{1,2,\ldots.n\}$ (for $k < \frac{n}{t+1}$), and to show that, somewhat surprisingly, all these results hold when the intersection requirement is replaced by a much weaker requirement. Other examples include stability versions of Frankl's recent result on the Erdős matching conjecture, the Ellis-Filmus-Friedgut proof of the Simonovits-Sós conjecture, and various EKR-type results on $r$-wise (cross)-$t$-intersecting families.

math.CO↗
arXiv2018

On the structure of subsets of the discrete cube with small edge boundary

The edge isoperimetric inequality in the discrete cube specifies, for each pair of integers $m$ and $n$, the minimum size $g_n(m)$ of the edge boundary of an $m$-element subset of $\{0,1\}^{n}$; the extremal families (up to automorphisms of the discrete cube) are initial segments of the lexicographic ordering on $\{0,1\}^n$. We show that for any $m$-element subset $\mathcal{F} \subset \{0,1\}^n$ and any integer $l$, if the edge boundary of $\mathcal{F}$ has size at most $g_n(m)+l$, then there exists an extremal family $\mathcal{G} \subset \{0,1\}^n$ such that $|\mathcal{F} Δ\mathcal{G}| \leq Cl$, where $C$ is an absolute constant. This is best-possible, up to the value of $C$. Our result can be seen as a `stability' version of the edge isoperimetric inequality in the discrete cube, and as a discrete analogue of the seminal stability result of Fusco, Maggi and Pratelli concerning the isoperimetric inequality in Euclidean space.

math.CO↗
arXiv2018

On set systems without a simplex-cluster and the Junta method

A family $\{A_{0},\ldots,A_{d}\}$ of $k$-element subsets of $[n]=\{1,2,\ldots,n\}$ is called a simplex-cluster if $A_{0}\cap\cdots\cap A_{d}=\varnothing$, $|A_{0}\cup\cdots\cup A_{d}|\le2k$, and the intersection of any $d$ of the sets in $\{A_{0},\ldots,A_{d}\}$ is nonempty. In 2006, Keevash and Mubayi conjectured that for any $d+1\le k\le\frac{d}{d+1}n$, the largest family of $k$-element subsets of $[n]$ that does not contain a simplex-cluster is the family of all $k$-subsets that contain a given element. We prove the conjecture for all $k\geζn$ for an arbitrarily small $ζ>0$, provided that $n\ge n_{0}(ζ,d)$. We call a family $\{A_{0},\ldots,A_{d}\}$ of $k$-element subsets of $[n]$ a $(d,k,s)$-cluster if $A_{0}\cap\cdots\cap A_{d}=\varnothing$ and $|A_{0}\cup\cdots\cup A_{d}|\le s$. We also show that for any $ζn\le k\le\frac{d}{d+1}n$ the largest family of $k$-element subsets of $[n]$ that does not contain a $(d,k,(\frac{d+1}{d}+ζ)k)$-cluster is again the family of all $k$-subsets that contain a given element, provided that $n\ge n_{0}(ζ,d)$. Our proof is based on the junta method for extremal combinatorics initiated by Dinur and Friedgut and further developed by Ellis, Keller, and the author.

math.CO↗
arXiv2018

Stability for the Complete Intersection Theorem, and the Forbidden Intersection Problem of Erdős and Sós

A family $F$ of sets is said to be $t$-intersecting if $|A \cap B| \geq t$ for any $A,B \in F$. The seminal Complete Intersection Theorem of Ahlswede and Khachatrian (1997) gives the maximal size $f(n,k,t)$ of a $t$-intersecting family of $k$-element subsets of $[n]=\{1,2,\ldots,n\}$, together with a characterisation of the extremal families. The forbidden intersection problem, posed by Erdős and Sós in 1971, asks for a determination of the maximal size $g(n,k,t)$ of a family $F$ of $k$-element subsets of $[n]$ such that $|A \cap B| \neq t-1$ for any $A,B \in F$. In this paper, we show that for any fixed $t \in \mathbb{N}$, if $o(n) \leq k \leq n/2-o(n)$, then $g(n,k,t)=f(n,k,t)$. In combination with prior results, this solves the above problem of Erdős and Sós for any constant $t$, except for in the ranges $n/2-o(n) < k < n/2+t/2$ and $k < 2t$. One key ingredient of the proof is the following sharp `stability' result for the Complete Intersection Theorem: if $k/n$ is bounded away from $0$ and $1/2$, and $F$ is a $t$-intersecting family of $k$-element subsets of $[n]$ such that $|F| \geq f(n,k,t) - O(\binom{n-d}{k})$, then there exists a family $G$ such that $G$ is extremal for the Complete Intersection Theorem, and $|F \setminus G| = O(\binom{n-d}{k-d})$. We believe this result to be of interest in its own right; indeed, it proves a conjecture of Friedgut from 2008. We prove it by combining classical `shifting' arguments with a `bootstrapping' method based upon an isoperimetric inequality. Another key ingredient is a `weak regularity lemma' for families of $k$-element subsets of $[n]$, where $k/n$ is bounded away from 0 and 1. This states that any such family $F$ is approximately contained within a `junta', such that the restriction of $F$ to each subcube determined by the junta is `pseudorandom' in a certain sense.

math.CO↗
arXiv2018

On a biased edge isoperimetric inequality for the discrete cube

The `full' edge isoperimetric inequality for the discrete cube (due to Harper, Bernstein, Lindsay and Hart) specifies the minimum size of the edge boundary $\partial A$ of a set $A \subset \{0,1\}^n$, as a function of $|A|$. A weaker (but more widely-used) lower bound is $|\partial A| \geq |A| \log_2(2^n/|A|)$, where equality holds iff $A$ is a subcube. In 2011, the first author obtained a sharp `stability' version of the latter result, proving that if $|\partial A| \leq |A| (\log(2^n/|A|)+ε)$, then there exists a subcube $C$ such that $|A ΔC|/|A| = O(ε/\log(1/ε))$. The `weak' version of the edge isoperimetric inequality has the following well-known generalization for the `$p$-biased' measure $μ_p$ on the discrete cube: if $p \leq 1/2$, or if $0 < p < 1$ and $A$ is monotone increasing, then $pμ_p(\partial A) \geq μ_p(A) \log_p(μ_p(A))$. In this paper, we prove a sharp stability version of the latter result, which generalizes the aforementioned result of the first author. Namely, we prove that if $pμ_p(\partial A) \leq μ_p(A) (\log_p(μ_p(A))+ε)$, then there exists a subcube $C$ such that $μ_p(A ΔC)/μ_p(A) = O(ε' /\log(1/ε'))$, where $ε' =ε\ln (1/p)$. This result is a central component in recent work of the authors proving sharp stability versions of a number of Erdős-Ko-Rado type theorems in extremal combinatorics, including the seminal `complete intersection theorem' of Ahlswede and Khachatrian. In addition, we prove a biased-measure analogue of the `full' edge isoperimetric inequality, for monotone increasing sets, and we observe that such an analogue does not hold for arbitrary sets, hence answering a question of Kalai. We use this result to give a new proof of the `full' edge isoperimetric inequality, one relying on the Kruskal-Katona theorem.

math.CO↗
arXiv2017

Approximation of biased Boolean functions of small total influence by DNF's

The influence of the $k$'th coordinate on a Boolean function $f:\{0,1\}^n \rightarrow \{0,1\}$ is the probability that flipping $x_k$ changes the value $f(x)$. The total influence $I(f)$ is the sum of influences of the coordinates. The well-known `Junta Theorem' of Friedgut (1998) asserts that if $I(f) \leq M$, then $f$ can be $ε$-approximated by a function that depends on $O(2^{M/ε})$ coordinates. Friedgut's theorem has a wide variety of applications in mathematics and theoretical computer science. For a biased function with $E[f]=μ$, the edge isoperimetric inequality on the cube implies that $I(f) \geq 2μ\log(1/μ)$. Kahn and Kalai (2006) asked, in the spirit of the Junta theorem, whether any $f$ such that $I(f)$ is within a constant factor of the minimum, can be $εμ$-approximated by a DNF of a `small' size (i.e., a union of a small number of sub-cubes). We answer the question by proving the following structure theorem: If $I(f) \leq 2μ(\log(1/μ)+M)$, then $f$ can be $εμ$-approximated by a DNF of size $2^{2^{O(M/ε)}}$. The dependence on $M$ is sharp up to the constant factor in the double exponent.

math.CO↗
arXiv2015

On the sum of the L1 influences of bounded functions

Let $f\colon \{-1,1\}^n \to [-1,1]$ have degree $d$ as a multilinear polynomial. It is well-known that the total influence of $f$ is at most $d$. Aaronson and Ambainis asked whether the total $L_1$ influence of $f$ can also be bounded as a function of $d$. Bačkurs and Bavarian answered this question in the affirmative, providing a bound of $O(d^3)$ for general functions and $O(d^2)$ for homogeneous functions. We improve on their results by providing a bound of $d^2$ for general functions and $O(d\log d)$ for homogeneous functions. In addition, we prove a bound of $d/(2 π)+o(d)$ for monotone functions, and provide a matching example.

cs.CC↗
arXiv2012

Monochromatic generating sets in groups and other algebraic structures

The \emph{generating chromatic number} of a group $G$, $\chigen(G)$, is the maximum number of colors $k$ such that there is a monochromatic generating set for each coloring of the elements of $G$ in $k$ colors. If no such maximal $k$ exists, we set $\chigen(G)=\infty$. Equivalently, $\chigen(G)$ is the maximal number $k$ such that there is no cover of $G$ by proper subgroups ($\infty$ if there is no such maximal $k$). We provide characterizations, for arbitrary gruops, in the cases $\chigen(G)=\infty$ and $\chigen(G)=2$. For nilpotent groups (in particular, for abelian ones), all possible chromatic numbers are characterized. Examples show that the characterization for nilpotent groups do not generalize to arbitrary solvable groups. We conclude with applications to vector spaces and fields.

math.GR↗