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Let $S_{N}(P)$ be the poset obtained by adding a dummy vertex on each diagonal edge of the $N$s of a finite poset $P$. We show that $S_{N}(S_{N}(P))$ is $N$-free. It follows that this poset is the smallest $N$-free barycentric subdivision of the diagram of $P$, poset whose existence was proved by P.A. Grillet. This is also the poset obtained by the algorithm starting with $P_0:=P$ and consisting at step $m$ of adding a dummy vertex on a diagonal edge of some $N$ in $P_m$, proving that the result of this algorithm does not depend upon the particular choice of the diagonal edge choosen at each step. These results are linked to drawing of posets.
In this note I provide two extensions of a particular case of the classical Poncelet theorem.
In this note, we prove a tight lower bound on the joint entropy of $n$ unbiased Bernoulli random variables which are $n/2$-wise independent. For general $k$-wise independence, we give new lower bounds by adapting Navon and Samorodnitskys Fourier proo
A hole is a chordless cycle with at least four vertices. A pan is a graph which consists of a hole and a single vertex with precisely one neighbor on the hole. An even hole is a hole with an even number of vertices. We prove that a (pan, even hole)-f
The class of all even-hole-free graphs has unbounded tree-width, as it contains all complete graphs. Recently, a class of (even-hole, $K_4$)-free graphs was constructed, that still has unbounded tree-width [Sintiari and Trotignon, 2019]. The class ha
We propose a new ternary infinite (even full-infinite) square-free sequence. The sequence is defined both by an iterative method and by a direct definition. Both definitions are analogous to those of the Thue-Morse sequence. The direct definition is