Gallai-Ramsey Numbers for C7 with Multiple Colors

The core idea of Ramsey theory is that complete disorder is impossible. Given a large structure, no matter how complex it is, we can always find a smaller substructure that has some sort of order. One view of this problem is in edge-colorings of complete graphs. For any graphs G, H1, ..., Hk, we write G → (H1, ..., Hk), or G → (H)k when H1 = ··· = Hk = H, if every k-edge-coloring of G contains a monochromatic Hi in color i for some i ∈ {1,...,k}. The Ramsey number rk(H1, ..., Hk) is the minimum integer n such that Kn → (H1, ..., Hk), where Kn is the complete graph on n vertices. Computing rk(H1, ..., Hk) is a notoriously difficult problem in combinatorics. A weakening of this problem is to restrict ourselves to Gallai colorings, that is, edge-colorings with no rainbow triangles. From this we define the Gallai-Ramsey number grk(K3,G) as the minimum integer n such that either Kn contains a rainbow triangle, or Kn → (G)k . In this thesis, we determine the Gallai-Ramsey numbers for C7 with multiple colors. We believe the method we developed can be applied to find grk(K3, C2n+1) for any integer n ≥ 2, where C2n+1 denotes a cycle on 2n + 1 vertices

The Study of Ramsey numbers r(C_k, C_k, C_k)

The Ramsey number r(C_k, C_k, C_k), denoted as r_3(C_k), is the smallest positive integer n such that any edge coloring with three colors of the complete graph on n vertices must contain at least one monochromatic cycle C_k. In this project, most literature on the Ramsey numbers r_3(C_k) are overviewed. Algorithms to check if a graph G contains any specific path or cycle and to construct extremal graphs for cycle C_k are developed. All good 3-colorings of complete graph K_10 are constructed to verify the value of Ramsey number r_3(C_4). Ramsey number value of r_3(C_3) is verified by direct point by point extension algorithm. The lower bounds for the Ramsey numbers r_3(C_5), r_3(C_6), and r_3(C_7) are provided as well. Additionally, the possibility of further research for larger k, especially for r_3(C_8) and r_3(C_10) is searched. Most of the results are based on computer algorithms

The Swimmer's view: does it really show what it is supposed to show? A retrospective study

<p>Abstract</p> <p>Background</p> <p>One of the basic principles in the primary survey of a trauma patient is immobilisation of the cervical spine till cleared of any injury. Lateral cervical spine radiograph is one of the important initial radiographic assessments. More than often additional radiographs like the Swimmer's view are necessary for adequate visualisation of the cervical spine. How good is the Swimmer's view in visualisation of the cervical spine after an inadequate lateral cervical spine radiograph?</p> <p>Methods</p> <p>100 Swimmer's view radiographs randomly selected over a 2 year period in trauma patients were included for the study. All the patients had inadequate lateral cervical spine radiographs. The radiographs were assessed with regards to their adequacy by a single observer. The criteria for adequacy were adequate visualisation of the C7 body, C7/T1 junction and the soft tissue shadow.</p> <p>Results</p> <p>Only 55% of the radiographs were adequate. None of the inadequate radiographs provided adequate visualisation of the C7 body and the C7/T1 junction. In 42.2% radiographs the soft tissue shadow was unclear. Poor exposure accounted for 53% of the inadequacies while overlapping bones accounted for the rest.</p> <p>Conclusion</p> <p>Clearing the cervical spine prior to removing triple immobilisation is essential in a trauma patient. This needs adequate visualisation from C1 to C7/T1 junction. In our study Swimmer's views did not satisfactorily provide adequate visualisation of the cervical spine in trauma patients. We recommend screening the cervical spine by a CT scan when the cervical spine lateral radiographs and Swimmer's views are inadequate.</p

Cervical stenosis in a professional rugby league football player: a case report

BACKGROUND: This paper describes a case of C7 radiculopathy in a professional rugby league player after repeated cervical spine trauma. The report outlines the management of the patient following an acute cervical hyperflexion injury with chiropractic manipulation and soft tissue therapies. It also presents a change in approach to include distractive techniques on presentation of a neurological deficit following re-injury. The clinical outcomes, while good, were very dependent upon the athlete restricting himself from further trauma during games, which is a challenge for a professional athlete. CASE PRESENTATION: A 30-year old male front row Australian rugby league player presented complaining of neck pain after a hyperflexion and compressive injury during a game. Repeated trauma over a four month period resulted in radicular pain. Radiographs revealed decreased disc height at the C5-C6 and C6-C7 levels and mild calcification within the anterior longitudinal ligament at the C6-C7 level. MRI revealed a right postero-lateral disc protrusion at the C6-C7 level causing a C7 nerve root compression. CONCLUSION: Recommendations from the available literature at the present time suggest that conservative management of cervical discogenic pain and disc protrusion, including chiropractic manipulation and ancillary therapies, can be successful in the absence of progressive neurological deficit. The current case highlights the initial successful management of a football athlete, and the later unsuccessful management. This case highlights the issues involvement in the management of a collision sport athlete with a serious neck injury