We prove that certain classes of metrically homogeneous graphs omitting triangles of odd short perimeter as well as triangles of long perimeter have the extension property for partial automorphisms and we describe their Ramsey expansions. This is extended variant of an extended abstract accepted to Eurocomb 2017 which contains proofs of the main statements.

Avocado crop (Persea americana Mill.) is of great commercial importance due to its high profitability. However, it being affected by various diseases and pests that affect yield reduce fruit quality. The aim this research was develop methodologies for the evaluation avocado plantations using different non-destructive technologies rapid phenotyping early detection incidence or damage stress in s...

The site-perimeter of a polyomino is the number of empty cells connected to the polyomino by an edge. A formula for the minimum site-perimeter with a given cell size is found. This formula is used to show the effectiveness of a simple random strategy in polyomino set achievement games.

We show that if the unit square is covered by n rectangles, then at least one must have perimeter at least 4(2m + 1)/(n + m(m + 1)), where m is the largest integer whose square is at most n. This result is exact for n of the form m(m + 1) (or m2).

We deal with the asymptotic behavior of the s-perimeter of a set E inside a domain Ω as s ↘ 0. We prove necessary and sufficient conditions for the existence of such limit, by also providing an explicit formulation in terms of the Lebesgue measure of E and Ω. Moreover, we construct examples of sets for which the limit does not exist.

An approach is presented for the enumeration of directed-convex polyominos that are not parallelogram polyominoes and we establish that there are ( 2n n−2 ) with a perimeter of 2n + 4. Finally using known results we prove that there are ( 2n n ) directed-convex polyominos with a perimeter of 2n + 4.

In the Heisenberg group, we prove that the boundary of sets with finite H-perimeter and having a bound on the measure theoretic normal is an H-Lipschitz graph. Then we show that if the normal is, on the boundary, the restriction of a continuous mapping, then the boundary is an H-regular surface.

In 1958 this was proved independently by S. Swierczkowski [15] and by P. Erdos and V.T. Sos [12],[13]. It is often called the Steinhaus theorem or Steinhaus Conjecture. A useful way of viewing the result is to think of a circle of unit circumference with points placed around the perimeter at distances 0, 0:, ... , (N 1)0: from an arbitrary origin on the perimeter. Then the distances between adj...

The planar soap bubble problem is the mathematically analogous problem in two dimensions to search for the least-perimeter way to enclose and separate regions R1, . . . , Rm of given areas A1, . . . , Am on the plane. In this work, we study the possible configurations for perimeter minimizing enclosures for more than three regions. In particular, we focus on the case of equal pressure regions. ...