Ramsey theory, a crucial theme in combinatorial theory, examines when arrangements or regularity could be discovered among sufficiently large and intricate combinatorial objects. The foundational belief stipulates that total disorder is non-viable in extensive systems, ensuring the emergence of monochromatic or highly structured subsets despite arbitrary divisions (Ramsey, 1930). Nonetheless, notwithstanding its profound theoretical impact, Ramsey theory has notable drawbacks when applied to substantial combinatorial structures. One notable limitation emerges from the avalanche-like escalation of Ramsey numbers, which gauge the least structural size needed to ensure certain regularities. Current Ramsey numbers develop exponentially or even faster, rendering pinpointing precise values exceedingly challenging, except for limited examples (Conlon, Fox, & Sudakov, 2015).

This combinatorial outgrowth adversely impacts practical applications, as the structural sizes required to ensure uniformity often exceed reasonable limits. Although Ramsey theory guarantees existence, it is predominantly unconstructive. It seldom presents explicit approaches to pinpointing the assured monochromatic subsets or configurations in sizable structures, hampering its efficacy for algorithmic or constructive purposes (Spencer, 1994). This nonconstructive nature complicates the translation of theoretical findings into efficient computational algorithms. The theory also grapples with difficulties as it extends classical findings to more intricate or higher-dimensional combinatorial objects.

For instance, multidimensional and hypergraph Ramsey problems are substantially more complex, and the known thresholds become more challenging to handle (Erdős & Hajnal, 1972). Consequently, the practicality and sharpness of Ramsey-type results diminish as the structures become more complex. Furthermore, probabilistic methods evince that typical extensive combinatorial structures might not show the extreme regularities assured by Ramsey theory until they become very large. It suggests a discrepancy between theoretical thresholds and natural behaviors (Alon & Spencer, 2016).

In summary, Ramsey theory effectively demonstrates the inevitability of an order in vast combinatorial structures, but at the same time, it has pitfalls in the exponential growth of Ramsey numbers, nonconstructive nature, increasing complexity in higher dimensions, and the disparity between theoretical thresholds and natural realities. The theory's shortfalls enthuse ongoing research to refine the thresholds and advance constructive approaches.

References:

Alon, N., & Spencer, J. H. (2016). The Probabilistic Method (4th ed.).

John Wiley & Sons. Conlon, D., Fox, J., & Sudakov, B. (2015). Recent developments in graph Ramsey theory. In Surveys in Combinatorics 2015(pp. 49-118). Cambridge University Press.

Erdős, P., & Hajnal, A. (1972). Unsolved problems in set theory. PRO-LSI. doi:10.1016/b978-0-444-10762-2.50006-1

Ramsey, F. P. (1930). On a problem of formal logic. London Mathematical Society, 30(4), 264-286. doi:10.1112/plms/s2-30.4.264

Spencer, J. (1994). Ten lectures on the probabilistic method. Society for Industrial and Applied Mathematics.