## What is/are Simple Combinatorial?

Simple Combinatorial - Certain conditions on $P$ imply that the expansions of $X_{inc(P)}$ in standard symmetric function bases yield coefficients which have simple combinatorial interpretations.^{[1]}We give a simple combinatorial description of an $(n-2k+2)$-chromatic edge-critical subgraph of the Schrijver graph $\mathrm{SG}(n,k)$, itself an induced vertex-critical subgraph of the Kneser graph $\mathrm{KG}(n,k)$.

^{[2]}The idea is simple: independently sample elements from the ground set and use simple combinatorial techniques (such as greedy or local search) on these sampled elements.

^{[3]}Among other results, (1) we give a simple combinatorial characterisation when a real forcing $\mathbb{P}_I$ can $+$-destroy a Borel ideal $\mathcal{J}$; (2) we discuss many classical examples of Borel ideals, their $+$-destructibility, and cardinal invariants; (3) we show that the Mathias-Prikry, $\mathbb{M}(\mathcal{I}^*)$-generic real $+$-destroys $\mathcal{I}$ iff $\mathbb{M}(\mathcal{I}^*)$ $+$-destroys $\mathcal{I}$ iff $\mathcal{I}$ can be $+$-destroyed iff $\mathrm{cov}^*(\mathcal{I},+)>\omega$; (4) we characterise when the Laver-Prikry, $\mathbb{L}(\mathcal{I}^*)$-generic real $+$-destroys $\mathcal{I}$, and in the case of P-ideals, when exactly $\mathbb{L}(\mathcal{I}^*)$ $+$-destroys $\mathcal{I}$; (5) we briefly discuss an even stronger form of destroying ideals closely related to the additivity of the null ideal.

^{[4]}An important role in the proof is played by analysis of subtle properties of a simple combinatorial object, an array of cyclic shifts of an arbitrary binary number.

^{[5]}One-dimensional eroders were studied by Gal’perin in the 1970s, who presented a simple combinatorial characterization of the class.

^{[6]}Using simple combinatorial methods, functional DNA transfer can be improved by up to four-fold of invaded cell populations.

^{[7]}Constructed through digital standard-cell delay chain and simple combinatorial logic, the module produces PWM signals with configurable on-time, period and time-delay (between phases) with resolution of a single delay-element.

^{[8]}This work provides a simple combinatorial strategy for the preparation of porous g-C3N4 nanosheets with low cost, environmental friendliness and enhanced photocatalytic activity.

^{[9]}One further feature of using the spectral independence approach to study colorings is that it avoids many of the technical complications in previous approaches caused by coupling arguments or by passing to the complex plane; the key improvement on the running time is based on relatively simple combinatorial arguments which are then translated into spectral bounds.

^{[10]}Here, we introduce a simple combinatorial approach that overcomes both extracellular and cellular barriers to enhance gene transfer efficacy in the lung in vivo.

^{[11]}In the centralized setting, we provide a surprisingly simple combinatorial algorithm that is asymptotically optimal in terms of both approximation factor and running time: an O(α)-approximation in linear time.

^{[12]}For this version we give a simple combinatorial approximation algorithm with ratio $5/3$, improving the previous $1.

^{[13]}We also present simple combinatorial relaxations based on computing maximum weighted matchings, which yield dual bounds towards finding minimum-weight fixed cardinality stable sets, and particular cases which are solvable in polynomial time.

^{[14]}) One of our key insights is to identify a simple combinatorial property of random XOR instances that makes spectral refutation work.

^{[15]}In this note, we describe a simple combinatorial way to view the power sum.

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