A frequency square is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only binary frequency squares of order $n$ with $n/2$ zeros and $n/2$ ones in each row and column. Two such frequency squares are orthogonal if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a set of $k$-MOFS$(n)$ is a set of $k$ binary frequency squares of order $n$ in which each pair of squares is orthogonal. A set of $k$-MOFS$(n)$ must satisfy $k\le(n-1)^2$, and any set of MOFS achieving this bound is said to be complete. For any $n$ for which there exists a Hadamard matrix of order $n$ we show that there exists at least $2^{n^2/4-O(n\log n)}$ isomorphism classes of complete sets of MOFS$(n)$. For $2<n\equiv2\pmod4$ we show that there exists a set of $17$-MOFS$(n)$ but no complete set of MOFS$(n)$. A set of $k$-maxMOFS$(n)$ is a set of $k$-MOFS$(n)$ that is not contained in any set of $(k+1)$-MOFS$(n)$. By computer enumeration, we establish that there exists a set of $k$-maxMOFS$(6)$ if and only if $k\in\{1,17\}$ or $5\le k\le 15$. We show that up to isomorphism there is a unique $1$-maxMOFS$(n)$ if $n\equiv2\pmod4$, whereas no $1$-maxMOFS$(n)$ exists for $n\equiv0\pmod4$. We also prove that there exists a set of $5$-maxMOFS$(n)$ for each order $n\equiv 2\pmod{4}$ where $n\geq 6$.

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Mubayi's Conjecture states that if F is a family of k-sized subsets of [n] = {1, . . . , n} which, for k ≥ d ≥ 3 and n ≥ dk d−1 , satisfies A 1 ∩· · ·∩A d = ∅ whenever |A 1 ∪· · ·∪A d | ≤ 2k for all distinct sets A 1 , . . . , A d ∈ F , then |F | ≤ n−1 k−1 , with equality occurring only if F is the family of all k-sized subsets containing some fixed element. This paper proves that Mubayi's Conjecture is true for all families that are invariant with respect to shifting; indeed, these families satisfy a stronger version of Mubayi's Conjecture. Relevant to the conjecture, we prove a fundamental bijective duality between what we call (i, j)unstable families and (j, i)-unstable families. Generalising previous intersecting conditions, we introduce the (d, s, t)-conditionally intersecting condition for families of sets and prove general results thereon. We prove fundamental theorems on two (d, s)-conditionally intersecting families that generalise previous intersecting families, and we pose an extension of a previous conjecture by Frankl and Füredi. Finally, we generalise a classical result by Erdős, Ko and Rado by proving tight upper bounds on the size of (2, s)-conditionally intersecting families F ⊆ 2 [n] and by characterising the families that attain these bounds. We extend this theorem for sufficiently large n to families F ⊆ 2 [n] whose members have at most a fixed size u.

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Alspach [Bull. Inst. Combin. Appl. 52 (2008), 7-20] defined the maximal matching sequencibility of a graph G, denoted ms(G), to be the largest integer s for which there is an ordering of the edges of G such that every s consecutive edges form a matching. In this paper, we consider the natural analogue for hypergraphs of this and related results and determine ms(λK n1,...,n k ) where λK n1,...,n k denotes the multi-k-partite k-graph with edge multiplicity λ and parts of sizes n 1 , . . . , n k , respectively. It turns out that these invariants may be given surprisingly precise and somewhat elegant descriptions, in a much more general setting.