A problem about “nine foreign journalists” from a Nordic Mathematical Olympiad is used as the starting point for a discussion of a class of extremal problems involving hypergraphs.Specifically, the problem is to find a sharp lower bound for the maximum degree of the hypergraph in terms of the number of (hyper)edges and their cardinalities.

In [Discrete Math. 111 (1993), 113-123], the \(c\)-th order edge toughness of a graph \(G\) is defined as
\[
\tau_c(G) = \min_{\substack{X \subseteq E(G), \&\omega(G – X) > c }} \left\{\frac{|X|}{\omega(G – X) – c}\right\},
\]
for any \(1 \leq c \leq |V(G)| – 1\). It is proved that \(\tau_c(G) \geq k\) if and only if \(G\) has \(k\) edge-disjoint spanning forests with exactly \(c\) components, and that for a given graph \(G\) with \(s = |E(G)|/(|V(G)| – c)\) and \(1 \leq c \leq |E(G)|\),
\(\tau_c(G) = s\) if and only if \(|E(H)| \leq s(|V(H)| – 1)\) for any subgraph \(H\) of \(G\). In this note, we shall present short proofs of the abovementioned theorems and shall indicate that these results can be extended to matroids.

In a group channel, codes correcting and detecting arbitrary patterns of errors (not necessarily “white noise”) are described metrically. This yields sphere-packing and Gilbert bounds on the sizes of all and of maximal codes, respectively. The loop transversal approach builds linear codes correcting arbitrary error patterns. In the binary case, the greedy loop transversal algorithm builds lexicodes.

A \(\lambda\)-design on \(v\) points is a family of \(v\) subsets (blocks) of a \(v\)-set such that any two distinct blocks intersect in \(\lambda\) points and not all blocks have the same cardinality.Ryser’s and Woodall’s \(\lambda\)-design conjecture states that each \(\lambda\)-design can be obtained from a symmetric design by complementing with respect to a fixed block. In a recent paper, we proved this conjecture for \(v = p+1, 2p+1, 3p+1\), where \(p\) is prime, and remarked that similar methods might work for \(v = 4p+1\). In the present paper, we prove the conjecture for \(\lambda\)-designs having replication numbers \(r\) and \(r^*\) such that \((r-1, r^*-1) = 4\) and, as a consequence, the \(\lambda\)-design conjecture is proved for \(v = 4p+1\), where \(p\) is prime.

In this paper, we obtain some combinatorial inequalities involving the parameters of a balanced array (B-array) \(T\) of strength four and with two levels. We discuss the usefulness of these inequalities in obtaining an upper bound for the number of constraints of \(T\), and briefly describe the importance of these arrays in the design of experiments as well as in combinatorics.

We call a partition \(\mu = (\mu_1, \ldots, \mu_k)\) of \(m\), \(m \leq n\), a constrained induced partition (cip) from a partition \(\lambda = (\lambda_1, \ldots, \lambda_r)\) of \(n\) if \(\mu_i \leq \lambda_i\) for \(i = 1, 2, \ldots, k\). In this paper, we study the set of cips (Sections 1-2), determine cips of size \(p\) (Section 4), and give a formula for the number of total subsequences with fixed size chosen from a given multiset such that the multiplicity of each digit in a subsequence is less than or equal to the multiplicity of this digit in the given multiset.

Let \(n \geq 2\) be an arbitrary integer. We show that for any two asymmetric digraphs \(D\) and \(F\) with \(m\)-\(\text{rad} F \geq \max\{4, n+1\}\), there exists an asymmetric digraph \(H\) such that \(m_M(H) \cong D\), \(m_P(H) \cong F\), and \(md(D, F) = n\).Furthermore, if \(K\) is a nonempty asymmetric digraph isomorphic to an induced subdigraph of both \(D\) and \(F\), then there exists a strong asymmetric digraph \(H\) such that \(m_M(H) \cong D\), \(m_P(H) \cong F\), and \(m_M(H) \cap m_P(H) \cong K\) if \(m\)-\(\text{rad}_{H_0}F \geq 4\), where \(H_0\) is a digraph obtained from \(D\) and \(F\) by identifying vertices similar to those in \(K\).

This paper addresses the following questions. In any graph \(G\) with at least \(\alpha\binom{n}{2}\) edges, how large of an induced subgraph \(H\) can we guarantee the existence of with minimum degree \(\delta(H) \geq \lfloor\alpha|V(H)|\rfloor\)? In any graph \(G\) with at least \(\alpha\binom{n}{2} – f(n)\) edges, where \(f(n)\) is an increasing function of \(n\), how large of an induced subgraph \(H\) can we guarantee the existence of containing at least \(\alpha\binom{|V(H)|}{2}\) edges? In any graph \(G\) with at least \(\alpha n^2\) edges, how large of an induced subgraph \(H\) can we guarantee the existence of with at least \(\alpha|V(H)|^2 + \Omega(n)\) edges? For \(\alpha = 1 – \frac{1}{r}\), for \(r = 2, 3, \ldots\), the answer is zero since if \(G\) is a complete \(r\)-partite graph, no subgraph \(H\) of \(G\) has more than \(\alpha|V(H)|^2\) edges. However, we show that for all admissible \(\alpha\) except these, the answer is \(\Omega(n)\). In any graph \(G\) with minimum degree \(\delta(G) \geq \alpha n – f(n)\), where \(f(n) = o(n)\), how large of an induced subgraph \(H\) can we guarantee the existence of with minimum degree \(\delta(H) \geq \Omega|V(H)|\)?

From any projective plane \(\Pi\) of even order \(n\) with an oval (\((n+2)\)-arc), a Hadamard \(3\)-design on \(n^2\) points can be defined using a well-known construction. If \(\Pi\) is Desarguesian with \(n = 2^m\) and the oval is regular (a conic plus nucleus) then it is shown that the binary code of the Hadamard \(3\)-design contains a copy of the first-order Reed-Muller code of length \(2^{2m}\).