Discrete mathematics

 

This task requires application of techniques 

based on special parts of mathematical theory. 

 

I don't have a sufficient knowledge of graph theory and discrete mathematics, so please help! You'll also probably need Ramseys numbers to solve this. I need to get the ABCDE and FGHIJ.

 

Very likely you have heard about Pooh's and Piglet's contributions to scenery improvement at Hundred Acre Wood. They have started with planting of simple and practical things, like haycorns and honeycombs, but with growing proficiency have shifted their gardening ambitions to more charming delights like mastershalum flowers. Although Piglet has expressed some doubts that these flowers probably might have been nasturtiums, of course that's mastershalums what they were, and the beauty of their flowerbed soon gained admiration by other inhabitants of Hundred Acre Wood.

 

Mastershalum planting is a very difficult thing unless you know how to do it

 

To follow up the story, this achievement gave also inspiration to others. Piglet soon decided to have his very own flower garden. He chose a species for which every single plant can simultaneously bloom many flowers of different colours and planted 18 seedlings. As it turned out, in his garden one could always find either 4 plants such that any pair of them had flowers of the same colour, or 4 plants such that no pair of them had flowers of the same colour.

 

Eeyore as usually was more sceptical about everything including gardening, so he tried it out in a more limited fashion. He planted just two plants and in his garden one could always find a pair of plants that either had or hadn't flowers of the same colour.

 

Inconspicuously wild blossoming of plants may not always be that innocuous as it could appear. It might be driven by some ancient order that only plants themselves know about.

 

Rabbit doubtlessly was very serious about all this and he decided to create a garden that would rival all other charms of Hundred Acre Wood. In his garden there should always be either 100 plants such that any pair of them had flowers of the same colour, or 100 plants such that no pair of them had flowers of the same colour. Knowing that such an undertaking could be a Very Serious Endeavour he thought it would be good to discuss this with Owl first. It certainly was a very resourceful idea, as the first thing that Owl said when hearing about Rabbit's plans was: "If you had not come to me, I should have come to you".

 

During their discussion Owl noted that Rabbit will need quite a few seedlings indeed – as everybody knows then (a long and boring number starting with digits ABCDE) would certainly be sufficient. However, for the Educated (who, for example, know that 18 plants for Piglet's garden are sufficient and that Piglet does not need 20) it should be obvious that also a smaller number of seedlings (another long and boring number, this one ending with digits FGHIJ) are enough. Some Very Educated (like Christopher Robin) probably could suggest even a smaller number, however, given Rabbit's enthusiasm and availability of help from his friends and relations, it probably will be faster for him to plant as many seedlings as Owl has said rather than to find out first exactly how many he might need. In a way, the more should be better, as Owl also remarked that, if Rabbit is having in mind the kind of plants that he thinks he is, he (Owl) would highly enjoy to have occasional invitations to have a nibble (or a puff) at a plant or few.

 

Hint: Wensleydale? Stilton? Fibonacci?