A graph is called well-covered if every maximal independent set has the same size. One generalization of independent sets in graphs is that of a fractional cover – attach nonnegative weights to the vertices and require that for every vertex the sum of all the weights in its closed neighbourhood be at least 1. In this paper, we consider and characterize fractionally well-covered graphs.

We prove that \(e(3,k+1,n) \geq 6n-13k\), where \(e(3,k+1,n)\) is the minimum number of edges in any triangle-free graph on \(n\) vertices with no independent set of size \(k+1\). To achieve this, we first characterize all such graphs with exactly \(e(3,k+1,n)\) edges for \(n \leq 3k\). These results yield some sharp lower bounds for the independence ratio for triangle-free graphs. In particular, the exact value of the minimal independence ratio for graphs with average degree \(4\) is shown to be \(\frac{4}{13}\). A slight improvement to the general upper bound for the classical Ramsey \(R(3,k)\) numbers is also obtained.

In this paper, we prove that for any \(n > 27363\), \(n \equiv 3\) modulo {6}, there exist a pair of orthogonal Steiner triple systems of order \(n\). Further, a pair of orthogonal Steiner triple systems of order \(n\) exist for all \(n \equiv 3\) modulo {6}, {3} \(< n \leq 27363\), with at most \(918\) possible exceptions. The proof of this result depends mainly on the construction of pairwise balanced designs having block sizes that are prime powers congruent to \(1\) modulo {6}, or \(15\) or \(27\). Some new examples are also constructed recursively by using conjugate orthogonal quasigroups.

We give a bijective proof for the identity \(S(n,k) \equiv \binom{n-j-1}{n-k} \pmod{2}\)
where \(j = \lfloor \frac{k}{2} \rfloor\) is the largest integer \(\leq\frac{k}{2}\) .

A bicover of pairs by quintuples of a \(v\)-set \(V\) is a family of 5-subsets of \(V\) (called blocks) with the property that every pair of distinct elements from \(V\) occurs in at least two blocks. If no other such bicover has fewer blocks, the bicover is said to be minimum, and the number of blocks in a minimum bicover is the covering number \(C_2(v, 5, 2)\), or simply \(C_2(v)\). It is well known that \(C_2(v) \geq \left \lceil \frac{v\left \lceil{(v-1)/2}\right \rceil}{5} \right \rceil = B_2(v)\), where \(\lceil x \rceil\) is the least integer not less than \(x\). It is shown here that if \(v\) is odd and \(v\not\equiv 3\) mod 10, \(v\not=9\) or 15,then \(C_2(v)=B(v)\).