Enumeration of three term arithmetic progressions in fixed density sets

Additive combinatorics is built around the famous theorem by Szemer\'edi which asserts existence of arithmetic progressions of any length among the integers. There exist several different proofs of the theorem based on very different techniques. Szemer\'edi's theorem is an existence statement, whereas the ultimate goal in combinatorics is always to make enumeration statements. In this article we develop new methods based on real algebraic geometry to obtain several quantitative statements on the number of arithmetic progressions in fixed density sets. We further discuss the possibility of a generalization of Szemer\'edi's theorem using methods from real algebraic geometry.Comment: 62 pages. Update v2: Corrected some references. Update v3: Incorporated feedbac

Rado Numbers and SAT Computations

Given a linear equation $\mathcal{E}$, the $k$-color Rado number $R_k(\mathcal{E})$ is the smallest integer $n$ such that every $k$-coloring of $\{1,2,3,\dots,n\}$ contains a monochromatic solution to $\mathcal E$. The degree of regularity of $\mathcal E$, denoted $dor(\mathcal E)$, is the largest value $k$ such that $R_k(\mathcal E)$ is finite. In this article we present new theoretical and computational results about the Rado numbers $R_3(\mathcal{E})$ and the degree of regularity of three-variable equations $\mathcal{E}$. % We use SAT solvers to compute many new values of the three-color Rado numbers $R_3(ax+by+cz = 0)$ for fixed integers $a,b,$ and $c$. We also give a SAT-based method to compute infinite families of these numbers. In particular, we show that the value of $R_3(x-y = (m-2) z)$ is equal to $m^3-m^2-m-1$ for $m\ge 3$. This resolves a conjecture of Myers and implies the conjecture that the generalized Schur numbers $S(m,3) = R_3(x_1+x_2 + \dots x_{m-1} = x_m)$ equal $m^3-m^2-m-1$ for $m\ge 3$. Our SAT solver computations, combined with our new combinatorial results, give improved bounds on $dor(ax+by = cz)$ and exact values for $1\le a,b,c\le 5$. We also give counterexamples to a conjecture of Golowich

Generalised Rado and Roth criteria

We study the Ramsey properties of equations $a_1P(x_1) + \cdots + a_sP(x_s) = b$, where $a_1,\ldots,a_s,b$ are integers, and $P$ is an integer polynomial of degree $d$. Provided there are at least $(1+o(1))d^2$ variables, we show that Rado's criterion and an intersectivity condition completely characterise which equations of this form admit monochromatic solutions with respect to an arbitrary finite colouring of the positive integers. Furthermore, we obtain a Roth-type theorem for these equations, showing that they admit non-constant solutions over any set of integers with positive upper density if and only if $b= a_1 + \cdots + a_s = 0$. In addition, we establish sharp asymptotic lower bounds for the number of monochromatic/dense solutions (supersaturation).Comment: 36 page

A new four parameter q-series identity and its partition implications

We prove a new four parameter q-hypergeometric series identity from which the three parameter key identity for the Goellnitz theorem due to Alladi, Andrews, and Gordon, follows as a special case by setting one of the parameters equal to 0. The new identity is equivalent to a four parameter partition theorem which extends the deep theorem of Goellnitz and thereby settles a problem raised by Andrews thirty years ago. Some consequences including a quadruple product extension of Jacobi's triple product identity, and prospects of future research are briefly discussed.Comment: 25 pages, in Sec. 3 Table 1 is added, discussion is added at the end of Sec. 5, minor stylistic changes, typos eliminated. To appear in Inventiones Mathematica

Distributed PCP Theorems for Hardness of Approximation in P

We present a new distributed model of probabilistically checkable proofs (PCP). A satisfying assignment $x \in \{0,1\}^n$ to a CNF formula $\varphi$ is shared between two parties, where Alice knows $x_1, \dots, x_{n/2}$, Bob knows $x_{n/2+1},\dots,x_n$, and both parties know $\varphi$. The goal is to have Alice and Bob jointly write a PCP that $x$ satisfies $\varphi$, while exchanging little or no information. Unfortunately, this model as-is does not allow for nontrivial query complexity. Instead, we focus on a non-deterministic variant, where the players are helped by Merlin, a third party who knows all of $x$. Using our framework, we obtain, for the first time, PCP-like reductions from the Strong Exponential Time Hypothesis (SETH) to approximation problems in P. In particular, under SETH we show that there are no truly-subquadratic approximation algorithms for Bichromatic Maximum Inner Product over {0,1}-vectors, Bichromatic LCS Closest Pair over permutations, Approximate Regular Expression Matching, and Diameter in Product Metric. All our inapproximability factors are nearly-tight. In particular, for the first two problems we obtain nearly-polynomial factors of $2^{(\log n)^{1-o(1)}}$; only $(1+o(1))$-factor lower bounds (under SETH) were known before