Joanna Boyland | Searcharxiv

arXiv2025

Big Ramsey Degrees of Countable Ordinals

Ramsey's theorem states that for all finite colorings of an infinite set, there exists an infinite homogeneous subset. What if we seek a homogeneous subset that is also order-equivalent to the original set? Let $S$ be a linearly ordered set and $a \in N$. The big Ramsey degree of $a$ in $S$, denoted $T(a,S)$, is the least integer $t$ such that, for any finite coloring of the $a$-subsets of $S$, there exists $S'\subseteq S$ such that (i) $S'$ is order-equivalent to $S$, and (ii) if the coloring is restricted to the $a$-subsets of $S'$ then at most $t$ colors are used. Mašulović \& Šobot (2019) showed that $T(a,ω+ω)=2^a$. From this one can obtain $T(a,ζ)=2^a$. We give a direct proof that $T(a,ζ)=2^a$. Mašulović and Šobot (2019) also showed that for all countable ordinals $α< ω^ω$, and for all $a \in N$, $T(a,α)$ is finite. We find exact value of $T(a,α)$ for all ordinals less than $ω^ω$ and all $a\in N$.

math.CO↗

arXiv2022

On sketching approximations for symmetric Boolean CSPs

A Boolean maximum constraint satisfaction problem, Max-CSP($f$), is specified by a predicate $f:\{-1,1\}^k\to\{0,1\}$. An $n$-variable instance of Max-CSP($f$) consists of a list of constraints, each of which applies $f$ to $k$ distinct literals drawn from the $n$ variables. For $k=2$, Chou, Golovnev, and Velusamy [CGV20, FOCS 2020] obtained explicit ratios characterizing the $\sqrt n$-space streaming approximability of every predicate. For $k \geq 3$, Chou, Golovnev, Sudan, and Velusamy [CGSV21, arXiv:2102.12351] proved a general dichotomy theorem for $\sqrt n$-space sketching algorithms: For every $f$, there exists $α(f)\in (0,1]$ such that for every $ε>0$, Max-CSP($f$) is $(α(f)-ε)$-approximable by an $O(\log n)$-space linear sketching algorithm, but $(α(f)+ε)$-approximation sketching algorithms require $Ω(\sqrt{n})$ space. In this work, we give closed-form expressions for the sketching approximation ratios of multiple families of symmetric Boolean functions. Letting $α'_k = 2^{-(k-1)} (1-k^{-2})^{(k-1)/2}$, we show that for odd $k \geq 3$, $α(k$AND$) = α'_k$, and for even $k \geq 2$, $α(k$AND$) = 2α'_{k+1}$. We also resolve the ratio for the "at-least-$(k-1)$-$1$'s" function for all even $k$; the "exactly-$\frac{k+1}2$-$1$'s" function for odd $k \in \{3,\ldots,51\}$; and fifteen other functions. We stress here that for general $f$, according to [CGSV21], closed-form expressions for $α(f)$ need not have existed a priori. Separately, for all threshold functions, we give optimal "bias-based" approximation algorithms generalizing [CGV20] while simplifying [CGSV21]. Finally, we investigate the $\sqrt n$-space streaming lower bounds in [CGSV21], and show that they are incomplete for $3$AND.

cs.DS↗