Two n × n Latin squares L 1 , L 2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that L 1 (i, j) = x and L 2 (i, j) = y. A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention.Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a (k + 1)-MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares.
Two $$n \times n$$ n × n Latin squares $$L_1, L_2$$ L 1 , L 2 are said to be orthogonal if, for every ordered pair (x, y) of symbols, there are coordinates (i, j) such that $$L_1(i,j) = x$$ L 1 ( i , j ) = x and $$L_2(i,j) = y$$ L 2 ( i , j ) = y . A k-MOLS is a sequence of k pairwise-orthogonal Latin squares, and the existence and enumeration of these objects has attracted a great deal of attention. Recent work of Keevash and Luria provides, for all fixed k, log-asymptotically tight bounds on the number of k-MOLS. To study the situation when k grows with n, we bound the number of ways a k-MOLS can be extended to a $$(k+1)$$ ( k + 1 ) -MOLS. These bounds are again tight for constant k, and allow us to deduce upper bounds on the total number of k-MOLS for all k. These bounds are close to tight even for k linear in n, and readily generalise to the broader class of gerechte designs, which include Sudoku squares.
Given any graph H, a graph G is said to be q-Ramsey for H if every coloring of the edges of G with q colors yields a monochromatic subgraph isomorphic to H. Further, such a graph G is said to be minimal q-Ramsey for H if additionally no proper subgraph G of G is q-Ramsey for H. In 1976, Burr, Erdős, and Lovász initiated the study of the parameter s q (H), defined as the smallest minimum degree among all minimal q-Ramsey graphs for H. In this paper, we consider the problem of determining how many vertices of degree s q (H) a minimal q-Ramsey graph for H can contain. Specifically, we seek to identify graphs for which a minimal q-Ramsey graph can contain arbitrarily many such vertices. We call a graph satisfying this property s q -abundant. Among other results, we prove that every cycle is s q -abundant for any integer q ≥ 2. We also discuss the cases when H is a clique or a clique with a pendant edge, extending previous results of Burr et al. and Fox et al. To prove our results and construct suitable minimal Ramsey graphs, we develop certain new gadget graphs, called pattern gadgets, which generalize and extend earlier constructions that have proven useful in the study of minimal Ramsey graphs. These new gadgets might be of independent interest.
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