Shi tableaux are special binary fillings of certain Young diagrams which arise in the study of Shi hyperplane arrangements related to classical root systems. For type A, the set T $\mathcal {T}$ of Shi tableaux naturally coincides with the set of Dyck paths, for which various notions of patterns have been introduced and studied over the years. In this paper we define a notion of pattern occurrence in T $\mathcal {T}$ which, although it can be regarded as a pattern on Dyck paths, it is motivated by the underlying geometric structure of the tableaux. Our main goal in this work is to study the poset of Shi tableaux defined by pattern-containment. More precisely, we determine explicit formulas for upper and lower covers for each T ∈ T $T\in \mathcal {T}$ , we consider pattern avoidance for the smallest non-trivial tableaux (size 2) and generalize these results to certain tableau of larger size. We conclude with open problems and possible future directions.

Kong and Ribemboim (1994) define for every poset P a sequence P = D0(P), D(P), D2(P), D3(P)… of posets, where Di(P) = D(Di− 1(P)) consists of all maximal antichains of Di− 1(P). They prove that for a finite poset P, there exists an integer i ≥ 0 such that Di(P) is a chain. In this paper, for every finite poset P, we show how to calculate the smallest integer i for which Di(P) is a chain.

We introduce the notion of a properly ordered coloring (POC) of a weighted graph, that generalizes the notion of vertex coloring of a graph. Under a POC, if xy is an edge, then the larger weighted vertex receives a larger color; in the case of equal weights of x and y, their colors must be different. In this paper, we shall initiate the study of this special coloring in graphs. For a graph G, we introduce the function f(G) which gives the maximum number of colors required by a POC over all weightings of G. We show that f(G) = ℓ(G), where ℓ(G) is the number of vertices of a longest path in G. Another function we introduce is χPOC(G; t) giving the minimum number of colors required over all weightings of G using t distinct weights. We show that the ratio of χPOC(G; t) − 1 to χ(G) − 1 can be bounded by t for any graph G; in fact, the result is shown by determining χPOC(G; t) when G is a complete multipartite graph. We also determine the minimum number of colors to give a POC on a vertex-weighted graph in terms of the number of vertices of a longest directed path in an orientation of the underlying graph. This extends the so called Gallai-Hasse-Roy-Vitaver theorem, a classical result concerning the relationship between the chromatic number of a graph G and the number of vertices of a longest directed path in an orientation of G.

We deal with first-order definability in the substructure ordering \(({\mathscr{D}}; \sqsubseteq )\) of finite directed graphs. In two papers, the author has already investigated the first-order language of the embeddability ordering \(({\mathscr{D}}; \leq )\). The latter has turned out to be quite strong, e.g., it has been shown that, modulo edge-reversing (on the whole graphs), it can express the full second-order language of directed graphs. Now we show that, with finitely many directed graphs added as constants, the first order language of \(({\mathscr{D}}; \sqsubseteq )\) can express that of \(({\mathscr{D}}; \leq )\). The limits of the expressive power of such languages are intimately related to the automorphism groups of the orderings. Previously, analogue investigations have found the concerning automorphism groups to be quite trivial, e.g., the automorphism group of \(({\mathscr{D}}; \leq )\) is isomorphic to \(\mathbb {Z}_{2}\). Here, unprecedentedly, this is not the case. Even though we conjecture that the automorphism group is isomorphic to \((\mathbb {Z}_{2}^{4} \times S_{4})\rtimes _{\alpha } \mathbb {Z}_{2}\), with a particular α in the semidirect product, we only prove it is finite.

Representing lattices by topologies has been studied to a great extent. In this paper, we prove that a complete lattice L is isomorphic to the lattice of open subsets of a P-space iff L is order generated by its countably prime elements. We establish the dual equivalence between the category of complete lattices order generated by their countably prime elements with morphisms preserving arbitrary sups and countable infs, and the category of countably sober P-spaces with morphisms of continuous maps. Finally, we show that the category of countably sober P-spaces is reflective in the category of P-spaces.

A subfamily {F_{1},F_{2},dots ,F_{|P|}}subseteq mathcal {F} of sets is a copy of a poset P in mathcal {F} if there exists a bijection phi :Prightarrow {F_{1},F_{2},dots ,F_{|P|}} such that whenever x le _{P} x^{prime } holds, then so does phi (x)subseteq phi (x^{prime }). For a family mathcal {F} of sets, let c(P,mathcal {F}) denote the number of copies of P in mathcal {F}, and we say that mathcal {F} is P-free if c(P,mathcal {F})=0 holds. For any two posets P, Q let us denote by La(n, P, Q) the maximum number of copies of Q over all P-free families mathcal {F} subseteq 2^{[n]}, i.e. max limits {c(Q,mathcal {F}): mathcal {F} subseteq 2^{[n]}, c(P,mathcal {F})=0 }. This generalizes the well-studied parameter La(n, P) = La(n, P, P1) where P1 is the one element poset, i.e. La(n, P) is the largest possible size of a P-free family. The quantity La(n, P) has been determined (precisely or asymptotically) for many posets P, and in all known cases an asymptotically best construction can be obtained by taking as many middle levels as possible without creating a copy of P. In this paper we consider the first instances of the problem of determining La(n, P, Q). We find its value when P and Q are small posets, like chains, forks, the N poset and diamonds. Already these special cases show that the extremal families are completely different from those in the original P-free cases: sometimes not middle or consecutive levels maximize La(n, P, Q) and sometimes the extremal family is not the union of levels. Finally, we determine (up to a polynomial factor) the maximum number of copies of complete multi-level posets in k-Sperner families. The main tools for this are the profile polytope method and two extremal set system problems that are of independent interest: we maximize the number of r-tuples A_{1},A_{2},dots , A_{r} in mathcal {A} over all antichains mathcal {A}subseteq 2^{[n]} such that (i) cap _{i=1}^{r}A_{i}=emptyset , (ii) cap _{i=1}^{r}A_{i}=emptyset and cup _{i=1}^{r}A_{i}=[n].

We investigate the possible values of the numbers of congruences of finite lattices of an arbitrary but fixed cardinality. Motivated by a result of Freese and continuing Czedli’s recent work, we prove that the third, fourth and fifth largest numbers of congruences of an n–element lattice are: 5 ⋅ 2n− 5 if n ≥ 5, 2n− 3 and 7 ⋅ 2n− 6 if n ≥ 6, respectively. We also determine the structures of the n–element lattices having 5 ⋅ 2n− 5, 2n− 3, respectively 7 ⋅ 2n− 6 congruences, along with the structures of their congruence lattices.

Let L be a finite n-element semilattice. We prove that if L has at least 127 ⋅ 2n− 8 subsemilattices, then L is planar. For n > 8, this result is sharp since there is a non-planar semilattice with exactly 127 ⋅ 2n− 8 − 1 subsemilattices.

Separation systems are posets with additional structure that form an abstract setting in which tangle-like clusters in graphs, matroids and other combinatorial structures can be expressed and studied. This paper offers some basic theory about infinite separation systems and how they relate to the finite separation systems they induce. They can be used to prove tangle-type duality theorems for infinite graphs and matroids, which will be done in future work that will build on this paper.

Let $n$ be a positive integer, $q$ a power of a prime, and $\mathcal{L}_n(q)$ the poset of subspaces of an $n$-dimensional vector space over a field with $q$ elements. This poset is a normalized matching poset and the set of subspaces of dimension $\left\lfloor n/2 \right\rfloor$ or those of dimension $\left\lceil n/2 \right\rceil$ are the only maximum-sized anti-chains in this poset. Strengthening this well-known and celebrated result, we show that, except in the case of $\mathcal{L}_3(2)$, these same collections of subspaces are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoid both a $\wedge$ and a $\vee$ as a subposet. We generalize some of the results to brooms and forks, and we also show that the union of the set of subspaces of dimension $k$ and $k+1$, for $k = \left\lfloor n/2 \right\rfloor$ or $k = \left\lceil n/2 \right\rceil-1$, are the only maximum-sized families in $\mathcal{L}_n(q)$ that avoid a butterfly (definitions below).