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Register a new userMaximal sets of mutually orthogonal frequency squares
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A frequency square is a square matrix in which each row and column is a permutation of the same multiset of symbols. A frequency square is of type $(n;lambda)$ if it contains $n/lambda$ symbols, each of which occurs $lambda$ times per row and $lambda$ times per column. In the case when $lambda=n/2$ we refer to the frequency square as binary. A set of $k$-MOFS$(n;lambda)$ is a set of $k$ frequency squares of type $(n;lambda)$ such that when any two of the frequency squares are superimposed, each possible ordered pair occurs equally often. A set of $k$-maxMOFS$(n;lambda)$ is a set of $k$-MOFS$(n;lambda)$ that is not contained in any set of $(k+1)$-MOFS$(n;lambda)$. For even $n$, let $mu(n)$ be the smallest $k$ such that there exists a set of $k$-maxMOFS$(n;n/2)$. It was shown in [Electron. J. Combin. 27(3) (2020), P3.7] that $mu(n)=1$ if $n/2$ is odd and $mu(n)>1$ if $n/2$ is even. Extending this result, we show that if $n/2$ is even, then $mu(n)>2$. Also, we show that whenever $n$ is divisible by a particular function of $k$, there does not exist a set of $k$-maxMOFS$(n;n/2)$ for any $kle k$. In particular, this means that $limsup mu(n)$ is unbounded. Nevertheless we can construct infinite families of maximal binary MOFS of fixed cardinality. More generally, let $q=p^u$ be a prime power and let $p^v$ be the highest power of $p$ that divides $n$. If $0le v-uh<u/2$ for $hge1$ then we show that there exists a set of $(q^h-1)^2/(q-1)$-maxMOFS$(n;n/q)$.
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A emph{frequency square} is a matrix in which each row and column is a permutation of the same multiset of symbols. We consider only {em binary} frequency squares of order $n$ with $n/2$ zeroes and $n/2$ ones in each row and column. Two such frequency squares are emph{orthogonal} if, when superimposed, each of the 4 possible ordered pairs of entries occurs equally often. In this context we say that a $k$-MOFS$(n)$ is a set of $k$ binary frequency squares of order $n$ in which each pair of squares is orthogonal. A $k$-MOFS$(n)$ must satisfy $kle(n-1)^2$, and any MOFS achieving this bound are said to be emph{complete}. For any $n$ for which there exists a Hadamard matrix of order $n$ we show that there exists at least $2^{n^2/4-O(nlog n)}$ isomorphism classes of complete MOFS$(n)$. For $2<nequiv2pmod4$ we show that there exists a $17$-MOFS$(n)$ but no complete MOFS$(n)$. A $k$-maxMOFS$(n)$ is a $k$-MOFS$(n)$ that is not contained in any $(k+1)$-MOFS$(n)$. By computer enumeration, we establish that there exists a $k$-maxMOFS$(6)$ if and only if $kin{1,17}$ or $5le kle 15$. We show that up to isomorphism there is a unique $1$-maxMOFS$(n)$ if $nequiv2pmod4$, whereas no $1$-maxMOFS$(n)$ exists for $nequiv0pmod4$. We also prove that there exists a $5$-maxMOFS$(n)$ for each order $nequiv 2pmod{4}$ where $ngeq 6$.
We give a simple construction of an orthogonal basis for the space of m by n matrices with row and column sums equal to zero. This vector space corresponds to the affine space naturally associated with the Birkhoff polytope, contingency tables and Latin squares. We also provide orthogonal bases for the spaces underlying magic squares and Sudoku boards. Our construction combines the outer (i.e., tensor or dyadic) product on vectors with certain rooted, vector-labeled, binary trees. Our bases naturally respect the decomposition of a vector space into centrosymmetric and skew-centrosymmetric pieces; the bases can be easily modified to respect the usual matrix symmetry and skew-symmetry as well.
We show that, in contrast to the integers setting, almost all even order abelian groups $G$ have exponentially fewer maximal sum-free sets than $2^{mu(G)/2}$, where $mu(G)$ denotes the size of a largest sum-free set in $G$. This confirms a conjecture of Balogh, Liu, Sharifzadeh and Treglown.
Nielsen proved that the maximum number of maximal independent sets (MISs) of size $k$ in an $n$-vertex graph is asymptotic to $(n/k)^k$, with the extremal construction a disjoint union of $k$ cliques with sizes as close to $n/k$ as possible. In this paper we study how many MISs of size $k$ an $n$-vertex graph $G$ can have if $G$ does not contain a clique $K_t$. We prove for all fixed $k$ and $t$ that there exist such graphs with $n^{lfloorfrac{(t-2)k}{t-1}rfloor-o(1)}$ MISs of size $k$ by utilizing recent work of Gowers and B. Janzer on a generalization of the Ruzsa-Szemeredi problem. We prove that this bound is essentially best possible for triangle-free graphs when $kle 4$.
A finite subset $X$ of the Euclidean space is called an $m$-distance set if the number of distances between two distinct points in $X$ is equal to $m$. An $m$-distance set $X$ is said to be maximal if any vector cannot be added to $X$ while maintaining the $m$-distance condition. We investigate a necessary and sufficient condition for vectors to be added to a regular simplex such that the set has only $2$ distances. We construct several $d$-dimensional maximal $2$-distance sets that contain a $d$-dimensional regular simplex. In particular, there exist infinitely many maximal non-spherical $2$-distance sets that contain both the regular simplex and the representation of a strongly resolvable design. The maximal $2$-distance set has size $2s^2(s+1)$, and the dimension is $d=(s-1)(s+1)^2-1$, where $s$ is a prime power.
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