6 research outputs found
Strong Ramsey Games in Unbounded Time
For two graphs and the strong Ramsey game on the
board and with target is played as follows. Two players alternately
claim edges of . The first player to build a copy of wins. If none of
the players win, the game is declared a draw. A notorious open question of Beck
asks whether the first player has a winning strategy in
in bounded time as . Surprisingly, in a recent paper Hefetz
et al. constructed a -uniform hypergraph for which they proved
that the first player does not have a winning strategy in
in bounded time. They naturally ask
whether the same result holds for graphs. In this paper we make further
progress in decreasing the rank.
In our first result, we construct a graph (in fact )
and prove that the first player does not have a winning strategy in
in bounded time. As an application of this
result we deduce our second result in which we construct a -uniform
hypergraph and prove that the first player does not have a winning
strategy in in bounded time. This improves the
result in the paper above.
An equivalent formulation of our first result is that the game
is a draw. Another reason for interest
on the board is a folklore result that the disjoint
union of two finite positional games both of which are first player wins is
also a first player win. An amusing corollary of our first result is that at
least one of the following two natural statements is false: (1) for every graph
, is a first player win; (2) for every graph
if is a first player win, then
is also a first player win.Comment: 18 pages, 46 figures; changes: fully reworked presentatio
Projected differences in isotopic ratios of N (Ξ΄<sup>15</sup>N in β°) between red blood cells and serum fractions in relation to gain or loss of body protein; the gradient of shading indicates the light (low Ξ΄<sup>15</sup>N) to heavy N (high Ξ΄<sup>15</sup>N).
<p>Projected differences in isotopic ratios of N (Ξ΄<sup>15</sup>N in β°) between red blood cells and serum fractions in relation to gain or loss of body protein; the gradient of shading indicates the light (low Ξ΄<sup>15</sup>N) to heavy N (high Ξ΄<sup>15</sup>N).</p
Sample sizes (<i>n</i>) of isotopic parameters measured in the blood of adult (β₯3 y) female caribou in Denali National Park and Preserve, Alaska.
a<p>The isotopic ratios of nitrogen (Ξ΄<sup>15</sup>N) in red blood cells.</p>b<p>Ξ΄<sup>15</sup>N in serum proteins.</p>c<p>Ξ΄<sup>15</sup>N in serum amino acids.</p>d<p>Difference between Ξ΄<sup>15</sup>N<sub>RBC</sub> and Ξ΄<sup>15</sup>N<sub>Proteins.</sub></p>e<p>Difference between Ξ΄<sup>15</sup>N<sub>RBC</sub> and Ξ΄<sup>15</sup>N<sub>AAs.</sub></p
A conceptual model of the routing of isotopes of N within a northern ungulate during winter.
<p>The size of each box indicates the relative size of each pool of N. The gradient of shading in each box indicates the range from less to more <sup>15</sup>N. Lighter arrows indicate flows of depleted N when animals are in positive N balance and gaining lean mass, while darker arrows indicate flows of enriched N when animals are losing lean mass.</p
The a) snowfall (22-year mean depicted by dashed line), b) size of the Denali Caribou Herd, 1986β2009 [40], c) birth masses of neonates [7], and d) body masses of adult females (β₯3 y) in late winter (L. Adams, USGS, unpublished data).
<p>Shading denotes winters with above average snowfall.</p
Winter and late winter locations of adult female caribou in Denali National Park and Preserve (Denali NPP), Alaska; blood was collected (<i>n</i>β=β168) for isotopic analyses at late winter locations during March 1993β2007.
<p>Winter and late winter locations of adult female caribou in Denali National Park and Preserve (Denali NPP), Alaska; blood was collected (<i>n</i>β=β168) for isotopic analyses at late winter locations during March 1993β2007.</p