Rainbow Ramsey Problems for the Boolean Lattice

We address the following rainbow Ramsey problem: For posets P, Q what is the smallest number n such that any coloring of the elements of the Boolean lattice Bn either admits a monochromatic copy of P or a rainbow copy of Q. We consider both weak and strong (non-induced and induced) versions of this problem.

Real subset sums and posets with an involution

In this paper, we carry out in an abstract order context some real subset combinatorial problems. Specifically, let [Formula: see text] be a finite poset, where [Formula: see text] is an order-reversing and involutive map such that [Formula: see text] for each [Formula: see text]. Let [Formula: see text] be the Boolean lattice with two elements and [Formula: see text] the family of all the order-preserving 2-valued maps [Formula: see text] such that [Formula: see text] if [Formula: see text] for all [Formula: see text]. In this paper, we build a family [Formula: see text] of particular subsets of [Formula: see text], that we call [Formula: see text]-bases on [Formula: see text], and we determine a bijection between the family [Formula: see text] and the family [Formula: see text]. In such a bijection, a [Formula: see text]-basis [Formula: see text] on [Formula: see text] corresponds to a map [Formula: see text] whose restriction of [Formula: see text] to [Formula: see text] is the smallest 2-valued partial map on [Formula: see text] which has [Formula: see text] as its unique extension in [Formula: see text]. Next we show how each [Formula: see text]-basis on [Formula: see text] becomes, in a particular context, a sub-system of a larger system of linear inequalities, whose compatibility implies the compatibility of the whole system.

Largest Family Without a Pair of Posets on Consecutive Levels of the Boolean Lattice

Suppose k ≥ 2 is an integer. Let Yk be the poset with elements x1,x2,y1,y2,…,yk− 1 such that y1 < y2 < ⋯ < yk− 1 < x1,x2 and let $Y_{k}^{\prime }$ Y k ′ be the same poset but all relations reversed. We say that a family of subsets of [n] contains a copy of Yk on consecutive levels if it contains k + 1 subsets F1,F2,G1,G2,…,Gk− 1 such that G1 ⊂ G2 ⊂⋯ ⊂ Gk− 1 ⊂ F1,F2 and |F1| = |F2| = |Gk− 1| + 1 = |Gk− 2| + 2 = ⋯ = |G1| + k − 1. If both Yk and $Y^{\prime }_{k}$ Y k ′ on consecutive levels are forbidden, the size of the largest such family is denoted by $\text {La}_{\mathrm {c}}\left (n, Y_{k}, Y^{\prime }_{k}\right )$ La c n , Y k , Y k ′ . In this paper, we will determine the exact value of $\text {La}_{\mathrm {c}}\left (n, Y_{k}, Y^{\prime }_{k}\right )$ La c n , Y k , Y k ′ .

De Finetti Lattices and Magog Triangles

The order ideal $B_{n,2}$ of the Boolean lattice $B_n$ consists of all subsets of size at most $2$. Let $F_{n,2}$ denote the poset refinement of $B_{n,2}$ induced by the rules: $i < j$ implies $\{i \} \prec \{ j \}$ and $\{i,k \} \prec \{j,k\}$. We give an elementary bijection from the set $\mathcal{F}_{n,2}$ of linear extensions of $F_{n,2}$ to the set of shifted standard Young tableau of shape $(n, n-1, \ldots, 1)$, which are counted by the strict-sense ballot numbers. We find a more surprising result when considering the set $\mathcal{F}_{n,2}^{1}$ of minimal poset refinements in which each singleton is comparable with all of the doubletons. We show that $\mathcal{F}_{n,2}^{1}$ is in bijection with magog triangles, and therefore is equinumerous with alternating sign matrices. We adopt our proof techniques to show that row reversal of an alternating sign matrix corresponds to a natural involution on gog triangles.

The Boolean Rainbow Ramsey Number of Antichains, Boolean Posets and Chains

Motivated by the paper, Boolean lattices: Ramsey properties and embeddings Order, 34 (2) (2017), of Axenovich and Walzer, we study the Ramsey-type problems on the Boolean lattices. Given posets $P$ and $Q$, we look for the smallest Boolean lattice $\mathcal{B}_N$ such that any coloring of elements of $\mathcal{B}_N$ must contain a monochromatic $P$ or a rainbow $Q$ as an induced subposet. This number $N$ is called the Boolean rainbow Ramsey number of $P$ and $Q$ in the paper. Particularly, we determine the exact values of the Boolean rainbow Ramsey number for $P$ and $Q$ being the antichains, the Boolean posets, or the chains. From these results, we also derive some general upper and lower bounds of the Boolean rainbow Ramsey number for general $P$ and $Q$ in terms of the poset parameters.

Improved Bounds for Induced Poset Saturation

Given a finite poset $\mathcal{P}$, a family $\mathcal{F}$ of elements in the Boolean lattice is induced-$\mathcal{P}$-saturated if $\mathcal{F}$ contains no copy of $\mathcal{P}$ as an induced subposet but every proper superset of $\mathcal{F}$ contains a copy of $\mathcal{P}$ as an induced subposet. The minimum size of an induced-$\mathcal{P}$-saturated family in the $n$-dimensional Boolean lattice, denoted $\mathrm{sat}^*(n,\mathcal{P})$, was first studied by Ferrara et al. (2017). Our work focuses on strengthening lower bounds. For the 4-point poset known as the diamond, we prove $\mathrm{sat}^*(n,\Diamond)\geq\sqrt{n}$, improving upon a logarithmic lower bound. For the antichain with $k+1$ elements, we prove $$\mathrm{sat}^*(n,\mathcal{A}_{k+1})\geq \left(1-\frac{1}{\log_2k}\right)\frac{kn}{\log_2 k}$$ for $n$ sufficiently large, improving upon a lower bound of $3n-1$ for $k\geq 3$.

An Efficient, Parallelized Algorithm for Optimal Conditional Entropy-Based Feature Selection

In Machine Learning, feature selection is an important step in classifier design. It consists of finding a subset of features that is optimum for a given cost function. One possibility to solve feature selection is to organize all possible feature subsets into a Boolean lattice and to exploit the fact that the costs of chains in that lattice describe U-shaped curves. Minimization of such cost function is known as the U-curve problem. Recently, a study proposed U-Curve Search (UCS), an optimal algorithm for that problem, which was successfully used for feature selection. However, despite of the algorithm optimality, the UCS required time in computational assays was exponential on the number of features. Here, we report that such scalability issue arises due to the fact that the U-curve problem is NP-hard. In the sequence, we introduce the Parallel U-Curve Search (PUCS), a new algorithm for the U-curve problem. In PUCS, we present a novel way to partition the search space into smaller Boolean lattices, thus rendering the algorithm highly parallelizable. We also provide computational assays with both synthetic data and Machine Learning datasets, where the PUCS performance was assessed against UCS and other golden standard algorithms in feature selection.