Strong Ramsey Games in Unbounded Time

For two graphs $B$ and $H$ the strong Ramsey game $\mathcal{R}(B,H)$ on the board $B$ and with target $H$ is played as follows. Two players alternately claim edges of $B$. The first player to build a copy of $H$ 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 $\mathcal{R}(K_n,K_k)$ in bounded time as $n\rightarrow\infty$. Surprisingly, in a recent paper Hefetz et al. constructed a $5$-uniform hypergraph $\mathcal{H}$ for which they proved that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(5)},\mathcal{H})$ 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 $G$ (in fact $G=K_6\setminus K_4$) and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n \sqcup K_n,G)$ in bounded time. As an application of this result we deduce our second result in which we construct a $4$-uniform hypergraph $G'$ and prove that the first player does not have a winning strategy in $\mathcal{R}(K_n^{(4)},G')$ in bounded time. This improves the result in the paper above. An equivalent formulation of our first result is that the game $\mathcal{R}(K_\omega\sqcup K_\omega,G)$ is a draw. Another reason for interest on the board $K_\omega\sqcup K_\omega$ 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 $H$, $\mathcal{R}(K_\omega,H)$ is a first player win; (2) for every graph $H$ if $\mathcal{R}(K_\omega,H)$ is a first player win, then $\mathcal{R}(K_\omega\sqcup K_\omega,H)$ is also a first player win.Comment: 18 pages, 46 figures; changes: fully reworked presentatio

Appendix A. Annual values of N content in organic matter at calving in fecal samples and composition of fecal samples (fecal nitrogen and ash content) collected from adult female migratory caribou (Rangifer tarandus) from the Rivière-George (RG) and Rivière-aux-Feuilles (RAF) herds, northern Quebec and Labrador, Canada.

Annual values of N content in organic matter at calving in fecal samples and composition of fecal samples (fecal nitrogen and ash content) collected from adult female migratory caribou (Rangifer tarandus) from the Rivière-George (RG) and Rivière-aux-Feuilles (RAF) herds, northern Quebec and Labrador, Canada

Appendix B. Details on the enrichment of 15N in maternal tissues, from blood clot to serum protein to muscle, of migratory caribou females (Rangifer tarandus) from the Rivière-George (RG) and Rivière-aux-Feuilles (RAF) herds, northern Quebec and Labrador, Canada.

Details on the enrichment of 15N in maternal tissues, from blood clot to serum protein to muscle, of migratory caribou females (Rangifer tarandus) from the Rivière-George (RG) and Rivière-aux-Feuilles (RAF) herds, northern Quebec and Labrador, Canada

Projected differences in isotopic ratios of N (δ¹⁵N in ‰) between red blood cells and serum fractions in relation to gain or loss of body protein; the gradient of shading indicates the light (low δ¹⁵N) to heavy N (high δ¹⁵N).

<p>Projected differences in isotopic ratios of N (δ¹⁵N in ‰) between red blood cells and serum fractions in relation to gain or loss of body protein; the gradient of shading indicates the light (low δ¹⁵N) to heavy N (high δ¹⁵N).</p

Sample sizes (<i>n</i>) of isotopic parameters measured in the blood of adult (≥3 y) female caribou in Denali National Park and Preserve, Alaska.

a<p>The isotopic ratios of nitrogen (δ¹⁵N) in red blood cells.</p>b<p>δ¹⁵N in serum proteins.</p>c<p>δ¹⁵N in serum amino acids.</p>d<p>Difference between δ¹⁵N_RBC and δ¹⁵N_Proteins.</p>e<p>Difference between δ¹⁵N_RBC and δ¹⁵N_AAs.</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 ¹⁵N. 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, 1986–2009 [40], c) birth masses of neonates [7], and d) body masses of adult females (≥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> = 168) for isotopic analyses at late winter locations during March 1993–2007.

<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> = 168) for isotopic analyses at late winter locations during March 1993–2007.</p