We present two results on maximal antichains in the strict chain product poset $[t_1+1]\times[t_2... more We present two results on maximal antichains in the strict chain product poset $[t_1+1]\times[t_2+1]\times\cdots\times[t_n+1]$. First, we prove that these maximal antichains are also maximum. Second, we prove that there is a bijection between maximal antichains in the strict chain product poset $[t_1+1]\times[t_2+1]\times\cdots\times[t_n+1]$ and antichains in the nonstrict chain product poset $[t_1]\times[t_2]\times\cdots\times[t_n]$.
We bound expected capture time and throttling number for the cop versus gambler game on a connect... more We bound expected capture time and throttling number for the cop versus gambler game on a connected graph with $n$ vertices, a variant of the cop versus robber game that is played in darkness, where the adversary hops between vertices using a fixed probability distribution. The paper that originally defined the cop versus gambler game focused on two versions, a known gambler whose distribution the cop knows, and an unknown gambler whose distribution is secret. We define a new version of the gambler where the cop makes a fixed number of observations before the lights go out and the game begins. We show that the strategy that gives the best possible expected capture time of $n$ for the known gambler can also be used to achieve nearly the same expected capture time against the observed gambler when the cop makes a sufficiently large number of observations. We also show that even with only a single observation, the cop is able to achieve an expected capture time of approximately $1.5n$,...
The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n... more The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally nonlinear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the number of ones and the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the upper bound on the number of ones in a minimally nonlinear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally nonlinear matrix and that there are not more than six ones in any other row of a minimally nonlinear matrix. Furthermore, we prove that if a minimally nonlinear 0-1 matrix has ones...
We present two results on maximal antichains in the strict chain product poset $[t_1+1]\times[t_2... more We present two results on maximal antichains in the strict chain product poset $[t_1+1]\times[t_2+1]\times\cdots\times[t_n+1]$. First, we prove that these maximal antichains are also maximum. Second, we prove that there is a bijection between maximal antichains in the strict chain product poset $[t_1+1]\times[t_2+1]\times\cdots\times[t_n+1]$ and antichains in the nonstrict chain product poset $[t_1]\times[t_2]\times\cdots\times[t_n]$.
We bound expected capture time and throttling number for the cop versus gambler game on a connect... more We bound expected capture time and throttling number for the cop versus gambler game on a connected graph with $n$ vertices, a variant of the cop versus robber game that is played in darkness, where the adversary hops between vertices using a fixed probability distribution. The paper that originally defined the cop versus gambler game focused on two versions, a known gambler whose distribution the cop knows, and an unknown gambler whose distribution is secret. We define a new version of the gambler where the cop makes a fixed number of observations before the lights go out and the game begins. We show that the strategy that gives the best possible expected capture time of $n$ for the known gambler can also be used to achieve nearly the same expected capture time against the observed gambler when the cop makes a sufficiently large number of observations. We also show that even with only a single observation, the cop is able to achieve an expected capture time of approximately $1.5n$,...
The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n... more The extremal function $ex(n, P)$ is the maximum possible number of ones in any 0-1 matrix with $n$ rows and $n$ columns that avoids $P$. A 0-1 matrix $P$ is called minimally nonlinear if $ex(n, P) = \omega(n)$ but $ex(n, P') = O(n)$ for every $P'$ that is contained in $P$ but not equal to $P$. Bounds on the number of ones and the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows were found in (CrowdMath, 2018). In this paper, we improve the upper bound on the number of ones in a minimally nonlinear 0-1 matrix with $k$ rows from $5k-3$ to $4k-4$. As a corollary, this improves the upper bound on the number of columns in a minimally nonlinear 0-1 matrix with $k$ rows from $4k-2$ to $4k-4$. We also prove that there are not more than four ones in the top and bottom rows of a minimally nonlinear matrix and that there are not more than six ones in any other row of a minimally nonlinear matrix. Furthermore, we prove that if a minimally nonlinear 0-1 matrix has ones...
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