kipriyanovichVertex Degree: The number of edges connected to a vertex in a graph. Ex: In a social network, your degree is how many friends you have. Pro-Tip: Sum of all degrees is always 2 * (num of edges) – the Handshaking Lemma!
Ross Kanghttps://arxiv.org/abs/2603.12358
Here is the *third* manuscript coming out of the "Topics in Ramsey theory" online-only problem-solving session (https://sparse-graphs.mimuw.edu.pl/doku.php?id=sessions:2025sessions:2025session1) of the Sparse (Graphs) Coalition, which took place less than a year ago.
It is still surprising to realise what one can make of such events, if they are set up well.
#combinatorics #remoteconferences #graphtheory #extremalcombinatorics #openscience
Ordered Ramsey and Turán numbers of alternating paths and their variants
An ordered graph is a graph whose vertex set is equipped with a total order. The ordered complete graph $K_N^<$ is the complete graph with vertex set $[N]$ equipped with the natural ordering of the integers. Given an ordered graph $H$, the ordered Ramsey number $R_<(H)$ is the smallest integer $N$ such that every red/blue edge-colouring of $K_N^<$ contains a monochromatic copy of $H$ with vertices appearing in the same relative order as in $H$. Balko, Cibulka, Král, and Kyn\v cl asked whether, among all ordered paths on $n$ vertices, the ordered Ramsey number is minimised by the alternating path $\mathrm{AP}_n$ -- the ordered path with vertex set $[n]$ such that the vertices encountered along the path are $1, n, 2, n - 1,3, n-2,\dots$. Motivated by this problem, we make progress on establishing the value of $R_<(\mathrm{AP}_n)$ by proving that \[ R_{<}(\mathrm{AP}_n)\leq \left(2+\frac{\sqrt{2}}{2}+o(1)\right)n. \] We then use similar methods to determine the exact ordered Turán number of $\mathrm{AP}_n$, and study the ordered Ramsey and Turán numbers of several related ordered paths.
Linotype Pilgrim
Peter CockI made another #ErgoMechKeyboard, taking the ribbon cable idea for my Bivouac34, and the 32-key layout and a 5-way navigation button from my Bivvy16D #SplitKeyboard. This is my Goldilocks32 #MechanicalKeyboard - another diode-free design with a #GraphTheory based sparse scanning matrix https://astrobeano.blogspot.com/2026/03/goldilocks-30-to-32-key-keyboard.html
Grant_HA #Higraph milestone: Blobs (nodes as sets) now work! Grab a 'parent' blob, and all the children move. Edges connect anywhere on the blob, and default to be orthogonal to the point of contact. graphML read and write working.
Now on to proper hyperedges!
Inquiry Into InquiryDifferential Logic • 18
If we follow the classical line which singles out linear functions as ideals of simplicity then we may complete the analytic series of the proposition in the following way.
The next venn diagram shows the differential proposition we get by extracting the linear approximation to the difference map at each cell or point of the universe What results is the logical analogue of what would ordinarily be called the differential of but since the adjective differential is being attached to just about everything in sight the alternative name tangent map is commonly used for whenever it’s necessary to single it out.
To be clear about what’s being indicated here, it’s a visual way of summarizing the following data.
To understand the extended interpretations, that is, the conjunctions of basic and differential features which are being indicated here, it may help to note the following equivalences.
Capping the analysis of the proposition in terms of succeeding orders of linear propositions, the final venn diagram of the series shows the remainder map which happens to be linear in pairs of variables.
Reading the arrows off the map produces the following data.
In short, is a constant field, having the value at each cell.
Resources
- Logic Syllabus
- Minimal Negation Operator
- Survey of Differential Logic
- Survey of Animated Logical Graphs
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
@haitchfiveI think this notation is starting to get somewhere now.
Let me assure you that there is real science behind this and it's not only pretty tetris things.
Differential Logic • 17
Enlargement and Difference Maps
Continuing with the example the following venn diagram shows the enlargement or shift map in the same style of field picture we drew for the tacit extension
A very important conceptual transition has just occurred here, almost tacitly, as it were. Generally speaking, having a set of mathematical objects of compatible types, in this case the two differential fields and both of the type is very useful, because it allows us to consider those fields as integral mathematical objects which can be operated on and combined in the ways we usually associate with algebras.
In the present case one notices the tacit extension and the enlargement are in a sense dual to each other. The tacit extension indicates all the arrows out of the region where is true and the enlargement indicates all the arrows into the region where is true. The only arc they have in common is the no‑change loop at If we add the two sets of arcs in mod 2 fashion then the loop of multiplicity 2 zeroes out, leaving the 6 arrows of shown in the following venn diagram.
Resources
- Logic Syllabus
- Minimal Negation Operator
- Survey of Differential Logic
- Survey of Animated Logical Graphs
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi
Differential Logic • 15
The structure of a differential field may be described as follows. With each point of there is associated an object of the following type: a proposition about changes in that is, a proposition In that frame of reference, if is the universe generated by the set of coordinate propositions then is the differential universe generated by the set of differential propositions The differential propositions and may thus be interpreted as indicating and respectively.
A differential operator of the first order type we are currently considering, takes a proposition and gives back a differential proposition In the field view of the scene, we see the proposition as a scalar field and we see the differential proposition as a vector field, specifically, a field of propositions about contemplated changes in
The field of changes produced by on is shown in the following venn diagram.
The differential field specifies the changes which need to be made from each point of in order to reach one of the models of the proposition that is, in order to satisfy the proposition
The field of changes produced by on is shown in the following venn diagram.
The differential field specifies the changes which need to be made from each point of in order to feel a change in the felt value of the field
Resources
- Logic Syllabus
- Minimal Negation Operator
- Survey of Differential Logic
- Survey of Animated Logical Graphs
cc: Academia.edu • Cybernetics • Laws of Form • Mathstodon (1) (2)
cc: Research Gate • Structural Modeling • Systems Science • Syscoi