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In [5] Graham and Rothschild consider a geometric Ramsey problem: finding the least n such that if all edges of the complete graph on the points {+1,-1}^n are 2-colored, there exist 4 coplanar points such that the 6 edges between them are monochromatic. They give an explicit upper bound: F(F(F(F(F(F(F(12))))))), where F(m) = 2^^(m)^^3, an extremely fast-growing function. By reducing the problem to a variant of the Hales-Jewett problem, we find an upper bound which is between F(4) and F(5).
We prove that there exists an absolute constant $delta>0$ such any binary code $Csubset{0,1}^N$ tolerating $(1/2-delta)N$ adversarial deletions must satisfy $|C|le 2^{text{poly}log N}$ and thus have rate asymptotically approaching 0. This is the firs
Let $lambda_{2}(G)$ be the second smallest normalized Laplacian eigenvalue of a graph $G$. In this paper, we determine all unicyclic graphs of order $ngeq21$ with $lambda_{2}(G)geq 1-frac{sqrt{6}}{3}$. Moreover, the unicyclic graphs with $lambda_{2}(G)=1-frac{sqrt{6}}{3}$ are also determined.
We describe a family of graphs with queue-number at most 4 but unbounded stack-number. This resolves open problems of Heath, Leighton and Rosenberg (1992) and Blankenship and Oporowski (1999).
Let ${rm ex}_{mathcal{P}}(n,T,H)$ denote the maximum number of copies of $T$ in an $n$-vertex planar graph which does not contain $H$ as a subgraph. When $T=K_2$, ${rm ex}_{mathcal{P}}(n,T,H)$ is the well studied function, the planar Turan number of
Reinhold et al. (Science, 1 May 2020, p. 518) provided two possible interpretations of measurements showing that the Sun is less active than other solar-like stars. We argue that one of those interpretations anticipates the observed differences betwe