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Suppose that a circular fire spreads in the plane at unit speed. A single fire fighter can build a barrier at speed v > 1. How large must v be to ensure that the fire can be contained, and how should the fire fighter proceed? We contribute two results. First, we analyze the natural curve FF v that develops when the fighter keeps building, at speed v , a barrier along the boundary of the expanding fire. We prove that the behavior of this spiralling curve is governed by a complex function (e w Z − s Z) − 1 , where w and s are real functions of v. For v > v c = 2. 6 1 4 4 ... all zeroes are complex conjugate pairs. If ϕ denotes the complex argument of the conjugate pair nearest to the origin then, by residue calculus, the fire fighter needs Θ (1 / ϕ) rounds before the fire is contained. As v decreases towards v c these two zeroes merge into a real one, so that argument ϕ goes to 0. Thus, curve FF v does not contain the fire if the fighter moves at speed v = v c . (That speed v > v c is sufficient for containing the fire has been proposed before by Bressan et al. [6], who constructed a sequence of logarithmic spiral segments that stay strictly away from the fire.) Second, we show that for any curve that visits the four coordinate half-axes in cyclic order, and in increasing distances from the origin the fire can not be contained if the speed v is less than 1.618..., the golden ratio. [ABSTRACT FROM AUTHOR]

In 2006, Alberto Bressan [1] suggested the following problem. Suppose a circular fire spreads in the Euclidean plane at unit speed. The task is to build, in real time, barrier curves to contain the fire. At each time t the total length of all barriers built so far must not exceed t ⋅ v , where v is a speed constant. How large a speed v is needed? He proved that speed v > 2 is sufficient, and that v > 1 is necessary. This gap of (1 , 2 ] is still open. The crucial question seems to be the following. When trying to contain a fire, should one build, at maximum speed, the enclosing barrier, or does it make sense to spend some time on placing extra delaying barriers in the fire's way? We study the situation where the fire must be contained in the upper L 1 half-plane by an infinite horizontal barrier to which vertical line segments may be attached as delaying barriers. Surprisingly, such delaying barriers are helpful when properly placed. We prove that speed v = 1.8772 is sufficient, while v > 1.66 is necessary. [ABSTRACT FROM AUTHOR]