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William Gasarch

Publications and source records attributed to William Gasarch.

34 records · Page 2Linked to original sources
arXiv2015

NIM with Cash

Let A be a finite subset of $\nat$. Then NIM(A;n) is the following 2-player game: initially there are $n$ stones on the board and the players alternate removing $a\in A$ stones. The first player who cannot move loses. This game has been well studied. We investigate an extension of the game where Player I starts out with d dollars, Player II starts out with e dollars, and when a player removes a\in A he loses a dollars. The first player who cannot move loses; however, note this can happen for two different reasons: (1) the number of stones is less than min(A), (2) the player has less than $\min(A)$ dollars. This game leads to more complex win conditions then standard NIM. We prove some general theorems from which we can obtain win conditions for a large variety of finite sets A. We then apply them to the sets A={1,L}, and A={1,L,L+1}.

math.CO↗
arXiv2015

Which Unbounded Protocol for Envy Free Cake Cutting is Better?

A division of a cake by n people is envy free if everyone thinks they got the biggest pieces. Note that peoples tastes can differ. There is a discrete protocol for envy free division for n=3 which takes at most 5 cuts. For n=4 and beyond there is a protocol but the number of cuts it takes is unbounded. In particular the number of cuts depends on peoples tastes. Given any number N peoples tastes can be set so that the algorithm takes over N cuts. There are three such algorithms known. Which is better? We have devised a way to measure the number of cuts even though it is unbounded. We use ordinals; therefore, a statement like "this protocol takes at most 2omega steps" makes sense. We analyse all three discrete algorithms for envy free cake cutting with this measure.

math.LO↗
arXiv2015

On the Sizes of DPDAs, PDAs, LBAs

There are languages A such that there is a Pushdown Automata (PDA) that recognizes A which is much smaller than any Deterministic Pushdown Automata (DPDA) that recognizes A. There are languages A such that there is a Linear Bounded Automata (Linear Space Turing Machine, henceforth LBA) that recognizes A which is much smaller than ny PDA that recognizes A. There are languages A such that both A and compliment(A) are recognizable by a PDA, but the PDA for A is much smaller than the PDA for compliment(A). There are languages A1, A2 such that A1,A2,A1 INTERSECT A_2 are recognizable by a PDA, but the PDA for A1 and A2 are much smaller than the PDA for A1 INTERSECT A2. We investigate these phenomenon and show that, in all these cases, the size difference is captured by a function whose Turing degree is on the second level of the arithmetic hierarchy. Our theorems lead to infinitely-often results. For example: for infinitely many $n$ there exists a language An recognized by a DPDA such that there is a small PDA for An, but any DPDA for An is large. We look at cases where we can get almost-all results, though with much smaller size differences.

cs.FL↗
arXiv2015

Distinct volume subsets

Suppose that $a$ and $d$ are positive integers with $a \geq 2$. Let $h_{a,d}(n)$ be the largest integer $t$ such that any set of $n$ points in $\mathbb{R}^d$ contains a subset of $t$ points for which all the non-zero volumes of the ${t \choose a}$ subsets of order $a$ are distinct. Beginning with Erdős in 1957, the function $h_{2,d}(n)$ has been closely studied and is known to be at least a power of $n$. We improve the best known bound for $h_{2,d}(n)$ and show that $h_{a,d}(n)$ is at least a power of $n$ for all $a$ and $d$.

math.CO↗
arXiv2015

Lower bounds on the Deterministic and Quantum Communication Complexity of Hamming Distance

Alice and Bob want to know if two strings of length n are almost equal. That is, do they differ on \textit{at most} a bits? Let 0\leq a\leq n-1. We show that any deterministic protocol, as well as any error-free quantum protocol (C* version), for this problem requires at least n-2 bits of communication. We show the same bounds for the problem of determining if two strings differ in exactly a bits. We also prove a lower bound of n/2-1 for error-free Q* quantum protocols. Our results are obtained by lower-bounding the ranks of the appropriate matrices.

cs.CC↗
arXiv2014

Proving programs terminate using well orderings, Ramsey Theory, and Matrices

Many programs allow the user to input data several times during its execution. If the program runs forever the user may input data infinitely often. A program terminates if it terminates no matter what the user does. We discuss various ways to prove that program terminates. The proofs use well orderings, Ramsey Theory, and Matrices. These techniques are used by real program checkers.

math.CO↗
arXiv2014

How many ways can you make change: Some easy proofs

Given a dollar, how many ways are there to make change using pennies, nickels, dimes, and quarters? What if you are given a different amount of money? What if you use different coin denominations? This is a well known problem and formulas are known. We present simpler proofs in several cases. We use recurrences to derive formulas if the coin denominations are {1,x,kx,rx}, and we use a simple proof using generating functions to derive a formula for any coin set.

math.HO↗
arXiv2013

Applications of the Canonical Ramsey Theorem to Geometry

Let P be a set of n points in R^d. How big is the largest subset X of P such that all of the distances determined between pairs are different? We show that X is at at least Omega(n^{1/6d}) This is not the best known; however the technique is new. Assume that no three of the original points are collinear. How big is the largest subset X of P such that all of the areas determined by elements of all triples are different? We show that, if d=2 then X is at least Omega((log log n)^{1/186}) and if d=3 then X is at least Omega((log log n)^{1/396}). We also obtain results for countable sets of points in R^d. All of our proofs use variants of the canonical Ramsey theorem and some geometric lemmas.

math.CO↗
arXiv2012

Rectangle Free Coloring of Grids

A two-dimensional \emph{grid} is a set $\Gnm = [n]\times[m]$. A grid $\Gnm$ is \emph{$c$-colorable} if there is a function $χ_{n,m}: \Gnm \to [c]$ such that there are no rectangles with all four corners the same color. We address the following question: for which values of $n$ and $m$ is $\Gnm$ $c$-colorable? This problem can be viewed as a bipartite Ramsey problem and is related to a the Gallai-Witt theorem (also called the multidimensioanl Van Der Waerden's Theorem). We determine (1) \emph{exactly} which grids are 2-colorable, (2) \emph{exactly} which grids are 3-colorable, and (3) \emph{exactly} which grids are 4-colorable. We use combinatorics, finite fields, and tournament graphs.

math.CO↗
arXiv2012

New Upper and Lower Bounds on the Rado Numbers

If E is a linear homogenous equation and c a natural then the Rado number $R_c(E)$ is the least N so that any c-coloring of the positive integers from 1 to N contains a monochromatic solution. Rado characterized for which E R_c(E) always exists. The original proof of Rado's theorem gave enormous bounds on R_c(E) (when it existed). In this paper we establish better upper bounds, and some lower bounds, for R_c(E) for some c and E. In the appendix we use some of our theorems, and ideas from a probabilistic SAT solver, to find many new Rado Numbers.

math.CO↗
arXiv2012

Three Proofs of the Hypergraph Ramsey Theorem (An exposition)

Ramsey, Erdos-Rado, and Conlon-Fox-Sudakov have given proofs of the 3-hypergraph Ramsey Theorem with better and better upper bounds on the 3-hypergraph Ramsey Number. Ramsey and Erdos-Rado also prove the a-hypergraph Ramsey Theorem. Conlon-Fox-Sudakov note that their upper bounds on the 3-hypergraph Ramsey Numbers, together with a recurrence of Erdos-Rado (which was the key to the Erdos-Rado proof), yield improved bounds on the a-hypergraph Ramsey numbers. We present all of these proofs and state explicit bounds for the 2-color case and the c-color case. We give a more detailed analysis of the construction of Conlon-Fox-Sudakov and hence obtain a slightly better bound.

math.CO↗
arXiv2012

A Statement in Combinatorics that is Independent of ZFC (an exposition)

It is known that, for any finite coloring of the naturals, there exists distinct naturals $e_1,e_2,e_3,e_4$ that are the same color such that $e_1+e_2=e_3+e_4$. Consider the following statement which we denote S: For every $\aleph_0$-coloring of the reals there exists distinct reals $e_1,e_2,e_3,e_4$ such that $e_1+e_2=e_3+e_4$?} Is it true? Erdos showed that S is equivalent to the negation of the Continuum Hypothesis, and hence S is indepedent of ZFC. We give an exposition of his proof and some modern observations about results of this sort.

math.CO↗
arXiv2011

Lower Bounds on the van der Waerden Numbers: Randomized- and Deterministic-Constructive

The van der Waerden number W(k,2) is the smallest integer n such that every 2-coloring of 1 to n has a monochromatic arithmetic progression of length k. The existence of such an n for any k is due to van der Waerden but known upper bounds on W(k,2) are enormous. Much effort was put into developing lower bounds on W(k,2). Most of these lower bound proofs employ the probabilistic method often in combination with the Lovász Local Lemma. While these proofs show the existence of a 2-coloring that has no monochromatic arithmetic progression of length k they provide no efficient algorithm to find such a coloring. These kind of proofs are often informally called nonconstructive in contrast to constructive proofs that provide an efficient algorithm. This paper clarifies these notions and gives definitions for deterministic- and randomized-constructive proofs as different types of constructive proofs. We then survey the literature on lower bounds on W(k,2) in this light. We show how known nonconstructive lower bound proofs based on the Lovász Local Lemma can be made randomized-constructive using the recent algorithms of Moser and Tardos. We also use a derandomization of Chandrasekaran, Goyal and Haeupler to transform these proofs into deterministic-constructive proofs. We provide greatly simplified and fully self-contained proofs and descriptions for these algorithms.

math.CO↗
arXiv2008

Square-Difference-Free Sets of Size Omega(n^{0.7334...})

A set A is square-difference free (henceforth SDF) if there do not exist x,y\in A, x\ne y, such that |x-y| is a square. Let sdf(n) be the size of the largest SDF subset of {1,...,n}. Ruzsa has shown that sdf(n) = Ω(n^{0.5(1+ \log_{65} 7)}) = Ω(n^{0.733077...}) We improve on the lower bound by showing sdf(n) = Ω(n^{0.5(1+ \log_{205} 12)})= Ω(n^{.7443...}) As a corollary we obtain a new lower bound on the quadratic van der Waerden numbers.

math.CO↗
arXiv1994

Reverse Mathematics and Recursive Graph Theory

We examine a number of results of infinite combinatorics using the techniques of reverse mathematics. Our results are inspired by similar results in recursive combinatorics. Theorems included concern colorings of graphs and bounded graphs, Euler paths, and Hamilton paths.

math.LO↗