# Combinatorics and Graph Theory

## 1801 Submissions

[3] **viXra:1801.0117 [pdf]**
*submitted on 2018-01-09 13:10:24*

### Cause/effect Correlations Through the Borsuk-Ulam Theorem and Kneser Graphs

**Authors:** Arturo Tozzi

**Comments:** 10 Pages.

The assessment of hidden causal relationships, e.g., adverse drug reactions in pharmacovigilance, is currently based on rather qualitative parameters. In order to find more quantifiable parameters able to establish the validity of the alleged correlations between drug intake and onset of symptoms, we introduce the Borsuk-Ulam Theorem (BUT), which states that a single point on a circumference projects to two points on a sphere. The BUT stands for a general principle that describes issues from neuroscience, theoretical physics, nanomaterials, computational topology, chaotic systems, group theory, cosmology. Here we introduce a novel BUT variant, termed operational-BUT, that evaluates causal relationships. Further, we demonstrate that the BUT is correlated with graph theory and in particular with the so-called Kneser graphs: this means that the combinatory features of observables, such as the bodily responses to drug intake, can be described in terms of dynamical mappings and paths taking place on well-established abstract structures. Therefore, physical and biological dynamical systems (including alleged causes and their unknown effects) make predictable moves into peculiar phase spaces, giving rise to
constrained trajectories that can be quantified.

**Category:** Combinatorics and Graph Theory

[2] **viXra:1801.0115 [pdf]**
*submitted on 2018-01-09 18:46:32*

### An Inquiry on the Implications of Abandoning Vixra and Using Arxiv on Finding the Shortest Path Within a Graph

**Authors:** Vixraisshit Usearxiv

**Comments:** 85 Pages. why are you still reading this? go to arXiv!!!

viXra is shitt
go use arXiv
if i can publish bs liek dis den why tf shood you trust anything on dis cite
P.S. I'm not illiterate, it's to prove my point.

**Category:** Combinatorics and Graph Theory

[1] **viXra:1801.0051 [pdf]**
*replaced on 2018-01-07 11:57:20*

### A Lemma on the Minimal Counter-example of Frankl's Conjecture

**Authors:** Ankush Hore

**Comments:** 5 Pages.

Frankl's Conjecture, from 1979, states that any finite union-closed family, containing at least one non-empty member set, must have an element which belongs to at least half of the member-sets. In this paper we list out some properties of the hypothetical minimal counter-example to this conjecture. In particular, we discuss the frequency of 3 distinct elements in the minimal counter-example. We also apply these findings to finite bipartite graphs.

**Category:** Combinatorics and Graph Theory