Thesis (Ph. D.)--West Virginia University, 2005. / Title from document title page. Document formatted into pages; contains vi, 76 p. Includes abstract. Includes bibliographical references (p. 75-76).

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem.

The LYM property of a finite standard graded poset is one of the central notions in Sperner theory. It is known that the product of two finite standard graded posets satisfying the LYM properties may not have the LYM property again. In 1974, Harper proved that if two finite standard graded posets satisfying the LYM properties also satisfy rank logarithmic concavities, then their product also satisfies these two properties. However, Harper's proof is rather non-intuitive. Giving a natural proof of Harper's theorem is one of the goals of this thesis. The main new result of this thesis is a characterization of rank-finite standard graded LYM posets that satisfy rank logarithmic concavities. With this characterization theorem, we are able to give a new, natural proof of Harper's theorem. In fact, we prove a strengthened version of Harper's theorem by weakening the finiteness condition to the rank-finiteness condition. We present some interesting applications of the main characterization theorem. We also give a brief history of Sperner theory, and summarize all the ingredients we need for the main theorem and its applications, including a new equivalent condition for the LYM property that is a key for proving our main theorem.

This thesis is primarily focused on problems in extremal combinatorics, although we will also consider some questions of analytic and algorithmic nature. The d-dimensional hypercube is the graph with vertex set {0,1}^d where two vertices are adjacent if they differ in exactly one coordinate. In Chapter 2 we obtain an upper bound on the 'saturation number' of Q_m in Q_d. Specifically, we show that for m ≥ 2 fixed and d large there exists a subgraph G of Q_d of bounded average degree such that G does not contain a copy of Q_m but, for every G' such that G ⊊ G' ⊆ Q_d, the graph G' contains a copy of Q_m. This result answers a question of Johnson and Pinto and is best possible up to a factor of O(m). In Chapter 3, we show that there exists ε > 0 such that for all k and for n sufficiently large there is a collection of at most 2^{(1-ε)k} subsets of [n] which does not contain a chain of length k+1 under inclusion and is maximal subject to this property. This disproves a conjecture of Gerbner, Keszegh, Lemons, Palmer, Pálvölgyi and Patkós. We also prove that there exists a constant c ∈ (0,1) such that the smallest such collection is of cardinality 2^{(1+o(1))^ck} for all k. In Chapter 4, we obtain an exact expression for the 'weak saturation number' of Q_m in Q_d. That is, we determine the minimum number of edges in a spanning subgraph G of Q_d such that the edges of E(Q_d)\E(G) can be added to G, one edge at a time, such that each new edge completes a copy of Q_m. This answers another question of Johnson and Pinto. We also obtain a more general result for the weak saturation of 'axis aligned' copies of a multidimensional grid in a larger grid. In the r-neighbour bootstrap process, one begins with a set A₀ of 'infected' vertices in a graph G and, at each step, a 'healthy' vertex becomes infected if it has at least r infected neighbours. If every vertex of G is eventually infected, then we say that A₀ percolates. In Chapter 5, we apply ideas from weak saturation to prove that, for fixed r ≥ 2, every percolating set in Q_d has cardinality at least (1+o(1))(d choose r-1)/r. This confirms a conjecture of Balogh and Bollobás and is asymptotically best possible. In addition, we determine the minimum cardinality exactly in the case r=3 (the minimum cardinality in the case r=2 was already known). In Chapter 6, we provide a framework for proving lower bounds on the number of comparable pairs in a subset S of a partially ordered set (poset) of prescribed size. We apply this framework to obtain an explicit bound of this type for the poset 𝒱(q,n) consisting of all subspaces of 𝔽_qⁿordered by inclusion which is best possible when S is not too large. In Chapter 7, we apply the result from Chapter 6 along with the recently developed 'container method,' to obtain an upper bound on the number of antichains in 𝒱(q,n) and a bound on the size of the largest antichain in a p-random subset of 𝒱(q,n) which holds with high probability for p in a certain range. In Chapter 8, we construct a 'finitely forcible graphon' W for which there exists a sequence (ε_i)^∞_i=1 tending to zero such that, for all i ≥ 1, every weak ε_i-regular partition of W has at least exp(ε_i^-2/2^{5log∗ε_i^-2}) parts. This result shows that the structure of a finitely forcible graphon can be much more complex than was anticipated in a paper of Lovász and Szegedy. For positive integers p,q with p/q ❘≥ 2, a circular (p,q)-colouring of a graph G is a mapping V(G) → ℤ_p such that any two adjacent vertices are mapped to elements of ℤ_p at distance at least q from one another. The reconfiguration problem for circular colourings asks, given two (p,q)-colourings f and g of G, is it possible to transform f into g by recolouring one vertex at a time so that every intermediate mapping is a p,q-colouring? In Chapter 9, we show that this question can be answered in polynomial time for 2 ≤ p/q < 4 and is PSPACE-complete for p/q ≥ 4.