We show that any connected Cayley graph $Gamma$ on an Abelian group of order $2n$ and degree $tilde{Omega}(log n)$ has at most $2^{n+1}(1 + o(1))$ independent sets. This bound is tight up to to the $o(1)$ term when $Gamma$ is bipartite. Our proof is
based on Sapozhenkos graph container method and uses the Pl{u}nnecke-Rusza-Petridis inequality from additive combinatorics.
By implementing algorithm
We determine the asymptotics of the number of independent sets of size $lfloor beta 2^{d-1} rfloor$ in the discrete hypercube $Q_d = {0,1}^d$ for any fixed $beta in [0,1]$ as $d to infty$, extending a result of Galvin for $beta in [1-1/sqrt{2},1]$. M
oreover, we prove a multivariate local central limit theorem for structural features of independent sets in $Q_d$ drawn according to the hard core model at any fixed fugacity $lambda>0$. In proving these results we develop several general tools for performing combinatorial enumeration using polymer models and the cluster expansion from statistical physics along with local central limit theorems.
We show that a random puncturing of a code with good distance is list recoverable beyond the Johnson bound. In particular, this implies that there are Reed-Solomon codes that are list recoverable beyond the Johnson bound. It was previously known that
there are Reed-Solomon codes that do not have this property. As an immediate corollary to our main theorem, we obtain better degree bounds on unbalanced expanders that come from Reed-Solomon codes.
Andreevs Problem states the following: Given an integer $d$ and a subset of $S subseteq mathbb{F}_q times mathbb{F}_q$, is there a polynomial $y = p(x)$ of degree at most $d$ such that for every $a in mathbb{F}_q$, $(a,p(a)) in S$? We show an $text{A
C}^0[oplus]$ lower bound for this problem. This problem appears to be similar to the list recovery problem for degree $d$-Reed-Solomon codes over $mathbb{F}_q$ which states the following: Given subsets $A_1,ldots,A_q$ of $mathbb{F}_q$, output all (if any) the Reed-Solomon codewords contained in $A_1times cdots times A_q$. For our purpose, we study this problem when $A_1, ldots, A_q$ are random subsets of a given size, which may be of independent interest.
Let $mathcal{H}$ be a $t$-regular hypergraph on $n$ vertices and $m$ edges. Let $M$ be the $m times n$ incidence matrix of $mathcal{H}$ and let us denote $lambda =max_{v perp overline{1},|v| = 1}|Mv|$. We show that the discrepancy of $mathcal{H}$ is
$O(sqrt{t} + lambda)$. As a corollary, this gives us that for every $t$, the discrepancy of a random $t$-regular hypergraph with $n$ vertices and $m geq n$ edges is almost surely $O(sqrt{t})$ as $n$ grows. The proof also gives a polynomial time algorithm that takes a hypergraph as input and outputs a coloring with the above guarantee.
We reprove the results on the hardness of approximating hypergraph coloring using a different technique based on bounds on the size of extremal $t$-agreeing families of $[q]^n$. Specifically, using theorems of Frankl-Tokushige [FT99], Ahlswede-Khacha
trian [AK98] and Frankl [F76] on the size of such families, we give simple and unified proofs of quasi NP-hardness of the following problems: $bullet$ coloring a $3$ colorable $4$-uniform hypergraph with $(log n)^delta$ many colors $bullet$ coloring a $3$ colorable $3$-uniform hypergraph with $tilde{O}(sqrt{log log n})$ many colors $bullet$ coloring a $2$ colorable $6$-uniform hypergraph with $(log n)^delta$ many colors $bullet$ coloring a $2$ colorable $4$-uniform hypergraph with $tilde{O}(sqrt{log log n})$ many colors where $n$ is the number of vertices of the hypergraph and $delta>0$ is a universal constant.
A covering code is a subset $mathcal{C} subseteq {0,1}^n$ with the property that any $z in {0,1}^n$ is close to some $c in mathcal{C}$ in Hamming distance. For every $epsilon,delta>0$, we show a construction of a family of codes with relative coverin
g radius $delta + epsilon$ and rate $1 - mathrm{H}(delta) $ with block length at most $exp(O((1/epsilon) log (1/epsilon)))$ for every $epsilon > 0$. This improves upon a folklore construction which only guaranteed codes of block length $exp(1/epsilon^2)$. The main idea behind this proof is to find a distribution on codes with relatively small support such that most of these codes have good covering properties.
We study hypergraph discrepancy in two closely related random models of hypergraphs on $n$ vertices and $m$ hyperedges. The first model, $mathcal{H}_1$, is when every vertex is present in exactly $t$ randomly chosen hyperedges. The premise of this is
closely tied to, and motivated by the Beck-Fiala conjecture. The second, perhaps more natural model, $mathcal{H}_2$, is when the entries of the $m times n$ incidence matrix is sampled in an i.i.d. fashion, each with probability $p$. We prove the following: 1. In $mathcal{H}_1$, when $log^{10}n ll t ll sqrt{n}$, and $m = n$, we show that the discrepancy of the hypergraph is almost surely at most $O(sqrt{t})$. This improves upon a result of Ezra and Lovett for this range of parameters. 2. In $mathcal{H}_2$, when $p= frac{1}{2}$, and $n = Omega(m log m)$, we show that the discrepancy is almost surely at most $1$. This answers an open problem of Hoberg and Rothvoss.
A rainbow $q$-coloring of a $k$-uniform hypergraph is a $q$-coloring of the vertex set such that every hyperedge contains all $q$ colors. We prove that given a rainbow $(k - 2lfloor sqrt{k}rfloor)$-colorable $k$-uniform hypergraph, it is NP-hard to
find a normal $2$-coloring. Previously, this was only known for rainbow $lfloor k/2 rfloor$-colorable hypergraphs (Guruswami and Lee, SODA 2015). We also study a generalization which we call rainbow $(q, p)$-coloring, defined as a coloring using $q$ colors such that every hyperedge contains at least $p$ colors. We prove that given a rainbow $(k - lfloor sqrt{kc} rfloor, k- lfloor3sqrt{kc} rfloor)$-colorable $k$ uniform hypergraph, it is NP-hard to find a normal $c$-coloring for any $c = o(k)$. The proof of our second result relies on two combinatorial theorems. One of the theorems was proved by Sarkaria (J. Comb. Theory. 1990) using topological methods and the other theorem we prove using a generalized Borsuk-Ulam theorem.
