We provide a method to construct \(t\)-designs from weighing matrices and association schemes. One instance of our method can produce a \(3\)-design from any (symmetric or skew-symmetric) conference matrix, thereby providing a partial answer to a question of Gunderson and Semeraro JCTB 2017. We explore variations of our method on some matrices that satisfy certain combinatorial restrictions. In particular, we show that there exist various infinite families of partially balanced incomplete block designs with block size four on the binary Hamming schemes and the \(3\)-class association schemes attached to symmetric designs, and regular pairwise balanced designs with block sizes three and four. Mathematics Subject Classifications: 05B05, 05C50, 05E30 Keywords: \(t\)-design, characteristic polynomial, weighing matrix, Hadamard matrix, association scheme
\enlargethispage{.2cm} If we two-colour a circle, we can always find an inscribed triangle with angles \((\frac{\pi}{7},\frac{2\pi}{7},\frac{4\pi}{7})\) whose three vertices have the same colour. In fact, Bialostocki and Nielsen showed that it is enough to consider the colours on the vertices of an inscribed heptagon. We prove that for every other triangle \(T\) there is a two-colouring of the circle without any monochromatic copy of \(T\). More generally, for \(k\geq 3\), call a \(k\)-tuple \((d_1,d_2,\dots,d_k)\) with \(d_1\geq d_2\geq \dots \geq d_k›0\) and \(\sum_{i=1}^k d_i=1\) a Ramsey \(k\)-tuple if the following is true: in every two-colouring of the circle of unit perimeter, there is a monochromatic \(k\)-tuple of points in which the distances of cyclically consecutive points, measured along the arcs, are \(d_1,d_2,\dots,d_k\) in some order. By a conjecture of Stromquist, if \(d_i=\frac{2^{k-i}}{2^k-1}\), then \((d_1,\dots,d_k)\) is Ramsey. Our main result is a proof of the converse of this conjecture. That is, we show that if \((d_1,\dots,d_k)\) is Ramsey, then \(d_i=\frac{2^{k-i}}{2^k-1}\). We do this by finding connections of the problem to certain questions from number theory about partitioning \(\mathbb{N}\) into so-called Beatty sequences. We also disprove a majority version of Stromquist's conjecture, study a robust version, and discuss a discrete version. Mathematics Subject Classifications: 05C15, 05C55, 11B75 Keywords: Ramsey theory, balanced sequences
This paper considers combinatorial games with the last move losing (mis & egrave;re play) instead of winning (normal play). Mis & egrave;re games form a pomonoid in which no nonzero elements are invertible; normal-play games, however, form a group with a much richer partial order. To compensate, mis & egrave;re researchers typically weaken the comparison relation by restricting to a subset of games, especially to a "universe" (closed under addition, conjugation, and options). For some well-studied universes (like the dicot and dead-ending universes), there exist finite comparison tests and also complete characterisations of their invertible elements; more recently, the invertible elements of every "parental" universe have been characterised. In this paper, we study the recently-defined universe of "blocking games" (containing both dicots and dead-ending games): we develop a finite comparison test, and we improve on the general invertibility characterisation by showing that, as in dead-ending, a game is invertible modulo blocking if and only if it is free of previous-win subpositions (or is equal to such a form).
We construct an uncountable family of well-quasi-ordered permutation classes, each with a distinct enumeration sequence. This disproves a conjecture that all well-quasi ordered permutation classes have algebraic generating functions, and in fact shows that many such classes lack D-finite or D-algebraic generating functions. Our construction is based on an uncountably large collection of factor-closed, well-quasi-ordered binary languages due to Pouzet.
Let n>2r>0 be integers. We consider families ℱ of subsets of an n-element set, in which the union of any two members has size at most 2r. One of our results states that for n≥ 6r the number of members of size exceeding r in ℱ is at most n-2r-1. Another result shows that for n>3.5r the number of sets of size at least r is at most nr. Both bounds are best possible and the latter sharpens the classical Katona Theorem. Similar results are proved for the odd case of the Katona Theorem as well.
