Let \(p(x > y)\) be the probability that a random linear extension of a finite poset has \(x\) above \(y\). Such a poset has a LEM (linear extension majority) cycle if there are distinct points \(x_1, x_2, \ldots, x_m\) in the poset such that \(p(x_1 > x_2) > \frac{1}{2}, p(x_2 > x_3) > \frac{1}{2}, \ldots, p(x_m > x_1) > \frac{1}{2}.\) We settle an open question by showing that interval orders can have LEM cycles.

We define the basis number, \(b(G)\), of a graph \(G\) to be the least integer \(k\) such that \(G\) has a \(k\)-fold basis for its cycle space. We investigate the basis number of the lexicographic product of paths, cycles, and wheels. It is proved that

\[b(P_n \otimes P_m) = b(P_n \otimes C_m) = 4 \quad \forall n,m \geq 7,\]
\[b(C_n \otimes P_m) = b(C_n \otimes C_m) = 4 \quad \forall n,m \geq 6,\]
\[b(P_n \otimes W_m) = 4 \quad \forall n,m \geq 9,\]
and
\[b(C_n \otimes W_n) = 4 \quad \forall n,m \geq 8.\]

It is also shown that \(\max \{4, b(G) + 2\}\) is an upper bound for \(b(P_n \otimes G)\) and \(b(C_n \otimes G)\) for every semi-hamiltonian graph \(G\).

Hare and Hare conjectured the 2-packing number of an \(m \times n\) grid graph to be \(\left\lceil \frac{mn}{5} \right\rceil\) for \(m, n \geq 9\). This is verified by finding the 2-packing number for grid graphs of all sizes.

We consider a subset-sum problem in \((2^\mathcal{S}, \cup)\), \((2^\mathcal{S}, \Delta)\), \((2^\mathcal{S}, \uplus)\), and \((\mathcal{S}_n, +)\), where \(S\) is an \(n\)-element set, \(\mathcal{S} \triangleq \{0,1,2,\ldots,2^n-1\}\), and \(\cup\), \(\Delta\), \(\uplus\), and \(+\) stand for set-union, symmetric set-difference, multiset-union, and real-number addition, respectively. Simple relationships between compatible pairs of sum-distinct sets in these structures are established. The behavior of a sequence \(\{n^{-1} |\mathcal{Z}| = 2, 3, \ldots\}\), where \(\mathcal{Z}\) is the maximum cardinality sum-distinct subset of \(\mathcal{S}\) (or \(\mathcal{S}_n\)), is described in each of the four structures.

Sixteen non-isomorphic symmetric \(2\)-\((31, 10, 3)\) designs with trivial full automorphism group are constructed.

We define a sequence of positive integers \({A} = (a_1, \ldots, a_n)\) to be a count-wheel of length \(n\) and weight \(w = a_1 + \cdots + a_n\) if it has the following property:
Let \(\overline{A}\) be the infinite sequence \((\overline{a_i})=(a_1, \ldots, a_n, a_1, \ldots, a_n, \ldots)\). Then there is a sequence \(0 = i(0) < i(1) < i(2) < \cdots\) such that for every positive integer \(k\), \(\overline{a}_{i(k-1)+1} + \cdots + \overline{a}_{i(k)} = k\). There are obvious notions of when a count-wheel is reduced or primitive. We show that for every positive integer \(w\), there is a unique reduced count-wheel of weight \(w\), denoted \([w]\). Also, \([w]\) is primitive if and only if \(w\) is odd. Further, we give several algorithms for constructing \([w]\), and a formula for its length. (Remark: The count-wheel \([15] = (1, 2, 3, 4, 3, 2)\) was discovered by medieval clock-makers.)

We present 3 connections between the two nonisomorphic \(C(6, 6, 1)\) designs and the exterior lines of an oval in the projective plane of order four. This connection demonstrates the existence of precisely four nonisomorphic large sets of \(C(6, 6, 1)\) designs.

Using computer algorithms we found that there exists a unique, up to isomorphism, graph on \(21\) points and \(125\) graphs on \(20\) points for the Ramsey number \(R(K_5 – e, K_5 – e) = 22\). We also construct all graphs on \(n\) points for the Ramsey number \(R(K_4 – e, K_5 – e) = 13\) for all \(n \leq 12\).

Affine \((\mu_1,\ldots,\mu_t)\)-resolvable \((\tau,\lambda)\)-designs are introduced. Constructions of such designs are presented.

Using basis reduction, we settle the existence problem for \(4\)-\((21,5,\lambda)\) designs with \(\lambda \in \{3,5,6,8\}\). These designs each have as an automorphism group the Frobenius group \(G\) of order \(171\) fixing two points. We also show that a \(4\)-\((21,5,1)\) design cannot have the subgroup of order \(57\) of \(G\) as an automorphism group.