Discrete Calculus Problem: Find the minimal value of (2008)
Aaron Robertson, Doron Zeilberger
Abstract: We prove that the minimum number (asymptotically) of monochromatic Schur triples that a 2-coloring of [1, n] can have is n2 22 + O(n). This revised version fills in a minor and subtle gap...
ON MONOCHROMATIC ASCENDING WAVES (2008)
Tim Lesaulnier, Aaron Robertson
A sequence of positive integers w1, w2,..., wn is called an ascending wave if wi+1 − wi ≥ wi − wi−1 for 2 ≤ i ≤ n − 1. For integers k, r ≥ 1, let AW (k; r) be the least positive...
We give some history and recent results in the area of pattern restricted permutations. We also present a new bijection between certain pattern restricted permutations.
Can Have, Aaron Robertson, Doron Zeilberger
: We prove the statement of the title, thereby solving a $100 problem of Ron Graham. This was solved independently by Tomasz Schoen. Tianjin, June 29, 1996: In a fascinating invited talk at the SOCA...
New Lower Bounds Formulas for Multicolored Ramsey (2007)
We give two lower bound formulas for multicolored Ramsey numbers. These formulas improve the bounds for several small multicolored Ramsey numbers. 1.
New Lower Bounds Formulas for Multicolored Ramsey (2007)
We give two lower bound formulas for multicolored Ramsey numbers. These formulas improve the bounds for several small multicolored Ramsey numbers. 1.
Some Two Color, Four Variable Rado Numbers (2007)
Robertson, Aaron, Myers, Kellen
There exists a minimum integer $N$ such that any 2-coloring of $\{1,2,...,N\}$ admits a monochromatic solution to $x+y+kz =\ell w$ for $k,\ell \in \mathbb{Z}^+$, where $N$ depends on $k$ and $\ell$....
Bounds on Van der Waerden Numbers and Some Related Functions (2007)
Brown, Tom, Landman, Bruce M., Robertson, Aaron
For positive integers $s$ and $k_1, k_2, ..., k_s$, let $w(k_1,k_2,...,k_s)$ be the minimum integer $n$ such that any $s$-coloring $\{1,2,...,n\} \to \{1,2,...,s\}$ admits a $k_i$-term arithmetic...
On the asymptotic minimum number of monochromatic 3-term arithmetic progressions (2006)
Parrilo, Pablo A., Robertson, Aaron, Saracino, Dan
Let V(n) be the minimum number of monochromatic 3-term arithmetic progressions in any 2-coloring of {1,2,...,n}. We show that (1675/32768) n^2 (1+o(1))
Two Color Off-diagonal Rado-type Numbers (2006)
Myers, Kellen, Robertson, Aaron
We show that for any two linear homogenous equations $\mathcal{E}_0,\mathcal{E}_1$, each with at least three variables and coefficients not all the same sign, any 2-coloring of $\mathbb{Z}^+$ admits...
On the degree of regularity of generalized van der Waerden triples (2005)
Frantzikinakis, Nikos, Landman, Bruce, Robertson, Aaron
Let $1 \leq a \leq b$ be integers. A triple of the form $(x,ax+d,bx+2d)$, where $x,d$ are positive integers is called an {\em (a,b)-triple}. The {\em degree of regularity} of the family of all...
Some New Exact van der Waerden Numbers (2005)
Landman, Bruce, Robertson, Aaron, Culver, Clay
For positive integers $r,k_0,k_1,...,k_{r-1},$ the van der Waerden number $w(k_0,k_1,...,k_{r-1})$ is the least positive integer $n$ such that whenever $\{1,2,...,n\}$ is partitioned into $r$ sets...
On Monochromatic Ascending Waves (2005)
LeSaulnier, Tim, Robertson, Aaron
A sequence of positive integers $w_1,w_2,...,w_n$ is called an ascending wave if $w_{i+1}-w_i \geq w_i - w_{i-1}$ for $2 \leq i \leq n-1$. For integers $k,r\geq1$, let $AW(k;r)$ be the least positive...
Avoiding Monochromatic Sequences With Special Gaps (2003)
Landman, Bruce M., Robertson, Aaron
For $S$ a set of positive integers, and $k$ and $r$ fixed positive integers, denote by $f(S,k;r)$ the least positive integer $n$ (if it exists) such that within every $r$-coloring of $\{1,2,...,n\}$...
Refined Restricted Involutions (2002)
Deutsch, Emeric, Robertson, Aaron, Saracino, Dan
Define $I_n^k(\alpha)$ to be the set of involutions of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_i$, for some $i \geq 2$, and define...
Refined Restricted Permutations Avoiding Subsets of Patterns of Length Three (2002)
Mansour, Toufik, Robertson, Aaron
Define $S_n^k(T)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid all patterns in $T \subseteq S_m$. We enumerate $S_n^k(T)$, $T \subseteq S_3$, for all...
Refined Restricted Permutations (2002)
Robertson, Aaron, Saracino, Dan, Zeilberger, Doron
Define $S_n^k(\alpha)$ to be the set of permutations of $\{1,2,...,n\}$ with exactly $k$ fixed points which avoid the pattern $\alpha \in S_m$. Let $s_n^k(\alpha)$ be the size of $S_n^k(\alpha)$. We...
New Lower Bound Formulas for Multicolored Ramsey Numbers (2001)
We give two lower bound formulas for multicolored Ramsey numbers. These formulas improve the bounds for several small multicolored Ramsey numbers.
Permutations Restricted by Two Distinct Patterns of Length Three (2000)
Define $S_n(R;T)$ to be the number of permutations on $n$ letters which avoid all patterns in the set $R$ and contain each pattern in the multiset $T$ exactly once. In this paper we enumerate...
On Generalized Van der Waerden Triples (1999)
Landman, Bruce, Robertson, Aaron
Van der Waerden's classical theorem on arithmetic progressions states that for any positive integers k and r, there exists a least positive integer, w(k,r), such that any r-coloring of...
Robertson, Aaron, Wilf, Herb, Zeilberger, Doron
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to...
Difference Ramsey Numbers and Issai Numbers (1999)
We present a recursive algorithm for finding good lower bounds for the classical Ramsey numbers. Using notions from this algorithm we then give some results for generalized Schur numbers, which we...
Permutations Containing and Avoiding 123 and 132 Patterns (1999)
We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern equals (n-2)2^(n-3). We then give a bijection onto the set of permutations which avoid 123-patterns...
The Number of Permutations With A Prescribed Number of 132 and 123 Patterns (1999)
Ekhad, Shalosh B., Robertson, Aaron, Zeilberger, Doron
Here we present the reasoning behind, and program to find, the generating functions for the number of permutations in the title. The article duals as the "accompanying" Maple package.
Permutation Patterns and Continued Fractions (1999)
Aaron Robertson, Herbert S. Wilf, Doron Zeilberger
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to...
Permutation Patterns and Continued Fractions (1999)
Aaron Robertson, Herbert S. Wilf, Doron Zeilberger
We find, in the form of a continued fraction, the generating function for the number of (132)-avoiding permutations that have a given number of (123) patterns, and show how to extend this to...
Permutations Containing and Avoiding 123 and 132 Patterns (1999)
this article we work towards the following generalization: How many permutations of length
Permutations Containing and Avoiding Patterns (1999)
this article we work towards the following generalization: How many permutations of length n avoid patterns p i , for i 0, and contain r j p j -patterns, for j 1, r j 1? We will first consider the...
New Lower Bounds for Some Multicolored Ramsey Numbers (1999)
In this article we use two different methods to find new lower bounds for some multicolored Ramsey numbers. In the first part we use the finite field method used by Greenwood and Gleason [GG] to show...
New Lower Bounds for Some Multicolored Ramsey Numbers (1999)
In this article we use two di#erent methods to find new lower bounds for some multicolored Ramsey numbers. In the first part we use the finite field method used by Greenwood and Gleason [GG] to show...
Permutations Containing and Avoiding 123and 132Patterns (1999)
We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern, equals (n-2)2 n-3, for n≥3. We then give a bijection onto the set of permutations which avoid...
New Lower Bounds for Some Multicolored Ramsey Numbers (1998)
We use finite fields and extend a result of Fan Chung to give eight new, nontrivial, lower bounds.
A 2-coloring of [1,n] can have (n^2)/22 + O(n) monochromatic Schur triples, but not less! (1998)
Robertson, Aaron, Zeilberger, Doron
We prove that the minimum number (asymptotically) of monochromatic Schur triples that a 2-coloring of [1,n] can have is (n^2)/22 + O(n). This was solved independently by Tomasz Schoen.
A 2-Coloring Of [1,N] Can Have (1/22)N² +O(N) MONOCHROMATIC SCHUR TRIPLES, BUT NOT LESS! (1998)
Aaron Robertson, Doron Zeilberger
: We prove the statement of the title, thereby solving a $100 problem of Ron Graham. This was solved independently by Tomasz Schoen. Tianjin, June 29, 1996: In a fascinating invited talk at the SOCA...
We prove that the number of permutations which avoid 132-patterns and have exactly one 123-pattern, equals (n